Possible that one dissertation in Computational Archaeology could earn two degrees?
It's hard to say something is definitely not possible, but I think this is a case where it just really isn't worthwhile. A PhD says you can do research; from there, it's more your research output that matters rather than the title of your degree.

What I would strongly suggest, however, is that you pursue having a mixture of faculty on your thesis committee. The exact rules will depend on your program and graduate school policies, but even if there is a requirement that official committee members come from a particular department, at a minimum you should be able to add some "outside" people who can help advise you in your second research area.

You should also pursue publishing papers in both fields.

Of course, all of this will depend on support from your main advisor, so when you look for particular programs, make that part of your search and part of the early conversations you have about your fit.