How should I remove the texture from the ceiling *without* tearing it out?

I was going to say "At least its not popcorn"!. You can also get a drywall sander that hooks up to a shop-vac to cut down on dust. I think I would do what Matthew suggests and see if it can be knocked down some with a scraper first. I would try a wall paper scraper, which I call a "razor blade on a stick". You can wet it to keep down dust, but I doubt it will soften it since its covered with 40 years of paint.

I have the same thing. We used two methods: 1st we used a sander to lower the high points of the texture, this is something you want to use as last resort. After the sanding was finished I did a skim coat, then had to sand again to smooth the skim coat. Super labor intensive. The next room I just put up 3/8 drywall, took half the time. It seems more practical to just give up a little space and hang drywall.

This kind of textured coatingis generally known as Artex in the UK (Wikipedia link.).

Take great care if you set to it with a sander as other answers have suggested, as old Artex can contain asbestos.

The two ways I've dealt with it previously have been to either skim over it or to remove it by first using a scraper to knock the tops off the peaks then to steam it with a wallpaper stripper and to scrape the Artex off the ceiling. Knocking the peaks off helps the steam penetrate through the layers of paint.

One coat of base coat Durabond 90 and one coat of regular 90 mud. First you have to knock down the high spots with a scraper. The 90 is a mud you have to mix yourself; it's made by Sheetrock brand.

The problem with re-hanging the drywall on the ceiling is the risk of getting halfway across the lid and finding that the framing no longer aligns with the joints. This could result in butt joints next to impossible to correctly finish. This has happened to me several times, forcing me to abort the re-hang and just skimming the texture already in place.

Scrape out only the high points with a floor scraper--don't go crazy and tear up the rock--then for speed use a drywall texture pump to spray an even coat of mud to cover old texture. Let this dry completely, then with a pole sander give it a light fast sand. Don't remove too much base, then spray texture again with thin mud on small dial on hopper. Give it a light orange peel.

This kind of coating is often put up to cover cracks etc in ceilings and is often not the original covering. This can give you an advantage. A steam wallpaper stripper can heat up the layers which will break the bonds of each coating and then it can be scraped/peeled off leaving the original plaster surface. You may need to sand down a spot first as a starting point.

I stumbled on a good method. I had a heavily textured ceiling with splattered mud. I soaked it and scraped it like all the advice I found online. It worked OK, but the best success came when I started using notched trowels to get the bulk of the work done. Do it in layers, soaking after each step...the first gets the surface scratched with a medium v-notch allowing water to soak in. next after another good soak use a hearty 1/4" square notch which sort of chisels it off. then fine square notch. Then clean up with a flat sheetrock trowel, or a flat masonry trowel works pretty good too. work at right angles so you make a waffle pattern. The idea is weaken it each step so it gives away without too much effort. Patience in soaking with spray bottle softens it quite nicely. The cieling underneath was painted underneath I should note...probably not the best method for bare sheetrock under.