How does the order of the pairs of cross-links in DNA determine the arrangement of the amino acids?

Quoting Richard Feynman from Chapter 3 of his book Six Easy Pieces, when he talks about DNA:

Attached to each sugar along the line, and linking the two chains together, are certain parts of cross-links. However, they are not all of the same kind; there are four kinds, called adenine, thymine, cytosine, and guanine, but let us call them A, B, C, and D.

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Next comes the question, precisely how does the order of the A, B, C, D units determine the arrangement of the amino acids in the protein? This is the central unsolved problem in biology today.

Feynman taught his lectures during 1961-1963. Do we now know how does the order of adenine, thymine, cytosine and guanine affect the order of the amino acids in the DNA?