Is "despatch" the British spelling for "dispatch" or is it an archaic spelling (or both)?

Despatch is still quite common in the UK. The manufacturing company I work for has a Despatch Team for example. In my view, in the UK despatch is traditionally used in preference to dispatch but that appears to be changing with younger people, who are more likely to use the American spelling.

Despatch can also be traced back to George Stephenson who named one of Rocket’s coaches Despatch along with Times and Experience. I always say dispatch. I’ve never heard anyone emphasise the E in despatch, even in this age of inflection

I would think that the spelling "despatch" is the more common one in the expression "Mentioned in despatches". At least that is what is used in the Australian War Memorial site, hence possibly reflecting Australian (and British) spelling.

Re name of George Stevenson's train (mentioned by Dave Maunder above) called "despatch", probably this has the connotation "fast" or "quickly", as in the expression "act with despatch", meaning "act quickly". I do not think that you would say "act with dispatch".

Ngram charts of 'despatch' versus 'dispatch'

The Ngram chart plotting the frequency of occurrence of "despatch" (blue line) versus "dispatch" (red line) for the period 1650–2019 is rather unusual:

The chart shows an extended period (roughly 160 years) of dominance for dispatch early on, followed by a sudden decline around 1810, accompanied by a rather remarkable rise for despatch—which, after peaking in 1864–1865, declines precipitously over the next two decades. The two spellings have remained (by historical standards) fairly close in frequency over the past 150 years, with dispatch enjoying an advantage since the middle 1930s.

Merriam-Webster's Eleventh Collegiate Dictionary (2003) gives despatch this brief treatment:

despatch chiefly Brit var of DISPATCH

Notably, the Eleventh Collegiate includes five distinct definitions of dispatch as a verb and another eight distinct definitions or subdefinitions of dispatch as a noun—and yet it implies that the variant spelling despatch may be used for any of these. As a quick check on the most basic application of the two spellings, I ran follow-up Ngram tests for "the despatch" versus "the dispatch" (to isolate noun use of the two forms) and "despatched" versus "dispatched" (to isolate verb use of the two). Here are the results for "the despatch" (blue line) versus "the dispatch" (red line):

And here are the results for "despatched" (blue line) versus "dispatched" (red line):

As these charts show, the paths of both variants in the noun form and the verb form I tested are roughly the same as the paths for the combined noun/verb form originally tested, meaning that when writers began referring to things as despatches they also began referring to despatching them.

The final test I ran with Ngram focused on usage of despatch and dispatch in specifically British published works. Here are the results for "despatch" (blue line) versus "dispatch" (red line) for the same 1650–2019 period:

Clearly, the preference for dispatch arrived much later in the UK than in the U.S.; and just as clearly, the orthographic tastes of British writers with regard to despatch/dispatch changed considerably over the course of the nineteenth century.

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Why did 'despatch' become popular?

The primary cause of the rise of despatch in Britain was evidently Samuel Johnson, A Dictionary of the English Language (1755), which offered entries for despatch as both a noun and a verb (along with multiple citations to instances where famous authors had used the word, spelled in every instance as despatch); only in stray quotations from earlier authors that arose in connection with the definitions of other words did Johnson use the spelling dispatch (presumably because that is the spelling those authors originally used).

In contrast both John Kersey, Dictionarium Anglo-Britannicum; or, A General English Dictionary, second edition (1715) and Nathan Bailey, An Universal Etymological Dictionary, second edition (1724) provide only the spelling dispatch in their entries. Perhaps Johnson was swayed in his spelling preference by his tracing of the English verb to the French word depescher, although Bailey had linked dispatch to depeché and despêcher without finding it desirable to import the e in place of the more frequent i.

Johnson's preference spilled over to early U.S. usage as well. Thus, despatch is the primary spelling given in Noah Webster, A Compendious History of the English Language (1806):

Despatch, v. t. to send away, finish, execute kill

Despatch, n. haste, speed, an express, management

...

Dispatch, see Despatch

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Webster's subsequent hostility to 'despatch'

In Webster's much larger and more detailed An American Dictionary of the English Language (1828), however, extensive definitions for the word as a transitive verb, an intransitive verb, and a noun appear in entries under the spelling dispatch, and the entry for despatch reads as follows:

DESPATCH. {See Dispatch.}

Conveniently, Webster addresses his rationale for this change of preference in the introduction to the 1828 American Dictionary:

In like manner [to the replacement by English lexicographers of sceptic with skeptic, which Webster deprecates], dispatch, which had, from time immemorial, been written with i, was changed into despatch, on the wonderful discovery,that the word was derived from the French depêcher. But why change one vowel and not the other? If we must follow the French, why not write despech, or depech? And why was this innovation limited to a single word? Why not carry the change through this whole class of words, and give us the benefit of uniformity? Is not disaster from the French desastre? Is not discharge from descharger? Is not disarm from desarmer? Is not disobey from desobeir? Is not disoblige from desobliger? Is not disorder from desordre? The prefix dis is more properly English than de, though both are used with propriety. But dispatch was the established orthography ; why then disturb the practice? Why select a single word from the whole class, and introduce a change which creates uncertainty where none had existed for ages, without the smallest benefit to indemnify us for the perplexity and discordance occasioned by the innovation?

It is gratifying to observe the stern good sense of the English nation, presenting a firm resistance to such innovations. Blackstone, Paley, Coxe, Milner, Scott and Mitford, uniformly use the old and genuine orthography of instructor, visitor, sceptic and dispatch.

Bold words from the same guy who, in the same dictionary, argued that systematize (the actual word in use) should be rendered as systemize for consistency with legalize, modernize, and civilize, and insisted that the "true spelling" of tongue is tung, of heinous is hainous, and of opaque is opake.

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The turn against 'despatch' in Britain

It is unclear to me when nineteenth-century British lexicographers began to turn away from despatch and toward dispatch, but by the time of Walter Skeat, A Concise Etymological Dictionary of the English Language, the main entry for the word had returned to dispatch:

Despatch; see Dispatch.

...

Dispatch, Despatch. (Span.–L.) Formerly spelt dis-, not des-. —Span. despachar, to dispatch, expedite. —L. dis- away; and L. type pactāre, to fasten, fix, from pactus, pp. of pangere, to fasten. ...

Henry Fowler, Modern English Usage (1926) voiced his support for dispatch in a brief entry:

dispatch, des-. The OED gives good reasons for preferring dis-.

Those "good reasons" appear in an introductory note to the entry for dispatch in The Compact Edition of the Oxford English Dictionary (1971):

Dispatch, despatch, v. ... {Found early in 16th c.: ad. It. dispacciare 'to dispatch, to hasten, to speed, to rid away any worke' (Florio), or Sp. despachar to expedite, to dispatch, to rid out of the way' (Minsheu). The radical is the same as in It. impacciare to entangle, hinder, stop, prevent, Sp. Pg. empachar to impede embarrass. Not related to F. depêcher, which gave the Engl. depesshe, DEPEACH, common in 15–16th c., rare after 1600, and app. superseded by dispatch before 1650. The uniform English spelling from the first introduction of the word to the early part of the 19th c. was with dis-; but in Johnson's Dictionary the word was somehow entered under des- (although Johnson himself always wrote dispatch, which is also the spelling of all the authors cited by him); though this has, since c 1820, introduced diversity into current usage, dispatch is to be preferred, as at once historical, and in accordance with English analogy; for even if the word had begun in ME with a form in des- from OF. (which it did not) , it would regularly have been spelt dis- by 1500.

As noted earlier in this answer, although it is true that the authors that Johnson cites in his thirteen examples of prior use of despatch originally spelled the word dispatch, he changed the spelling in each instance (in that entry, anyway) to despatch. It is therefore highly implausible that the word "was somehow entered under des-" by some sort of mischance, without Johnson's clear intention to adopt that spelling.

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Conclusions

The spelling despatch seems to have taken hold in British English mainly on the strength of Samuel Johnson's use of the spelling in his highly influential Dictionary of the English Language (1755). That isn't to say that previous English writers never used the spelling. Early English Books Online finds 86 matches for various forms of despatch, from as early as a 1542 translation of Antoine Geuffroy, The Order of the Greate Turckes Courte, of Hys Menne of Warre, and of All Hys Conquestes, with the Summe of Mahumetes Doctryne:

He [the great Turk] heareth gladly messengers and embassadoures straungers, chyefely yf they brynge hym presentes. But he practyseth nothing with them, but only heareth them declare the cause of theyr cōming. To whiche he answereth nothyng or yf he answere, he sayth onely, I haue hearde the, resorte to the Baches, and they shall despatche the, after this they se hym no more.

But EEBO also finds 59 matches for various forms of dyspatch from as early as a 1523 translation of Froissart's Chronicles and 11,349 matches for various forms of dispatch from at least as early as a 1525 translation of Froissart. It thus seems clear that dispatch was far and away the mot common spelling of the word before Johnson intervened.

If we may judge from Noah Webster's first dictionary (from 1806), U.S. English began with a preference for despatch, inherited from Johnson; but Webster began to reject that spelling in favor of dispatch in 1828, and U.S. orthography seems to have made the switch earlier than British orthography did. By the end of the nineteenth century, British lexicographers—most prominently, Charles Murray of Oxford's New Dictionary project—had reappraised the alternatives and come out in favor of dispatch.

Given the OED's endorsement of dispatch near the close of the nineteenth century, the lingering (but diminishing) preference for despatch in British publications over the course of most of the twentieth century may perhaps be best understood as a tribute to orthographic inertia. Since the turn of the twenty-first century, however, dispatch has become more frequent than despatch in British publications overall.

To Dispatch and despatch are two different things. Dispatch is to finish, assign or complete a task. To Despatch is to send or deliver. A Despatch Department in a company sends out mail, both internally and internally.

Source Grammarist.com