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.

You will need to check with the university/universities where you plan to submit your PhD thesis. Before you ask anybody in person, I suggest to check the examination guidelines first. I found a couple of negative examples in a quick Google search:

"A doctoral thesis that was already submitted to another faculty may not be the basis of the doctoral examination." Uni Bonn

"A student may not submit for assessment thesis material which has already been submitted for another degree at Sultan Qaboos University or any other academic institution." Sultan Qaboos University

"The application to become a doctoral candidate shall be rejected if the doctoral committee finds that ... the subject of the candidate's dissertation has already been submitted successfully in another subject." Uni Frankfurt

Before I answer, I'll say this. As someone with a PhD in archaeology...certainly get your PhD in CS. If for some reason later on you found yourself wanting to apply for a job in an archaeology department, a PhD in CS focused on archaeology is perfectly acceptable, depending on the position.

To answer your question in a round about way, I wonder if your idea of a PhD is just a little bit inaccurate. The PhD is a bar you pass, a bit of proof that you are able to do a certain level of analysis that leads to a (very small) development in one or more academic fields. A single PhD can contribute to one field or it could contribute to several fields, but you don't get a separate PhD for every field you make a contribution to. A lot of research is multidisciplinary, but that doesn't warrant a PhD in every discipline that you're working in. You get a PhD from the (almost always) one field that the bulk of your method+theory rests in, in the department who agrees to accept and examine your work. Further, this is the department that you or some funder has been paying for the years of your PhD.

Take me for example. I got a PhD in Archaeology. Aside from a terrifying viva comment that was basically "We think this is a PhD, just maybe not an Archaeology PhD", all was fine. However my research was more in Criminology and Politics, with a bit of Law, and since then I've had a strong academic career...though I've never been in an archaeology department again. I do, however, know how to lay down a tight 1x1m square and always know where my trowel is.

Here's something you may not have considered: would an archaeology department accept you? If you don't have a background or previous degrees in archaeology, and don't know the full background of archaeological method and theory, it doesn't matter if your project helps archaeology, you likely aren't going to make it past the admission hurdle. Now perhaps you were some sort of double major, okay, but take a good look at your project. Is it a CS project that just uses some aspect of archaeology as an example, test case, or data set? If so, there isn't an archaeology PhD in that.

Another thought: does your project actually require archaeology supervision? Will your CS department be able to advise you on the archaeological aspects of your project. At my Uni, if this really was a crossover project, we would suggest you have an archaeologist on your supervisory team. You'd still only get the one PhD of course, but you'd have someone who was there to make sure what you were doing wouldn't be considered insane in archaeological circles.

Lastly, some advice. It looks like you're thinking that it can't hurt to ask about getting two PhDs for one project. It might not hurt if you don't plan on having ties with archaeology but, really, they are going to think you're a bit mad for asking. At best, it sounds naive about, again, what a PhD is. Better would be to introduce yourself and your project and ask about collaboration or supervision arrangements.