Ph.D. in engineering after Ph.D. in mathematics I want to start soon with a Ph.D. in mathematics, though I'm not sure, that long-term I want to stay in mathematics. But I know that I'm going to like doing mathematics full-time right now and invest the effort required for a Ph.D. so this unsureness will not deter me. Additionally, because I'm quite old, I want to do the Ph.D. fast(er that usual).
[This remark is bound to generate comments, I know - as "old" is relative, assume that I am compared to my colleagues and that I am bothered by the fact that I'm lagging behind compared to them; speeding up my Ph.D. is reasonable I believe, because 
1) being "older" I've had more time to make sure I have all the mathematical prerequisits needed for the subfield I want to get my Ph.D. in (e.g. where some of my colleagues learned about Crandall-Rabinowitz only after being admitted to a Ph.D program, I know about it already now. "Bang. Just like that.")
2) I want to do the Ph.D in my home university (which is in Europe), where I already know all the relevant professors, have already taken seminars with them, know which Ph.D. programs exist, how I can get into them, how the system works etc. so I won't loose time by these non-mathematical things and I'm less hindered by formal criteria.]

Ok, this sound all nice and dandy, so where's the question. Here: Given the facts, that
 a) I may not want to stay professionally in mathematics in the long-term, a Ph.D. degree from a very famous U.S. university, as they're world-wide the most well-known, could prove helpful for getting a job in the industry in a Western country later on (of course this doesn't apply to France, were you'd better have studied at ENS and not at Harvard, but I think I'm safe to chose a country were a degree from e.g. Harvard weighs) and that
 b) I also like engineering applications of mathematics and I've read for myself books in a certain area of engineering and started taking occasional related, non-mathematical courses there and like to see research in this area,
I may like to apply, after having obtained by Ph.D. degree in mathematics, at a famous U.S. university for a second Ph.D. degree in engineering. (MIT comes to mind as prototypical example.)

What effect does a previous Ph.D. (in mathematics) has for me getting accepted at a highly ranked U.S. university ? 
(Good, bad, irrelevant ? It will show that I can to research, which is good, but not that good, since its research in mathematics and not engineering. If I have to do one of those pesky standarized tests, like GRE (do you know good programs, where this may be waived a Ph.D. ?), does this weigh more then the Ph.D. ?

Further reflections:    


Besides the reputation of the university of the top U.S. universities, it seems that in the United States formal admission criteria for Ph.D. programs are less strict as in Europe (for an extreme example, take a look at the careers of these professors), so not having a bachelor in engineering may not knock me out from the start.)
For the nitpickers: Note that actually I'm not getting a Ph.D. in mathematics but a doctoral degree.
I also considered the possibility of directly applying for a Ph.D. in engineering in the U.S. But I doubt that I'll be able to do it as fast as the one in mathematics, since none of the reasons from 1) and 2) apply. Additionally I may have to exclude some programs, because my prerequisits aren't sufficient. So there only seem to be disadvantages to doing one directly.
From what I know, it would hurt you. From the MIT ECE website (just as an example):

If I already have a PhD, can I apply for another PhD in EECS?
No, we will not admit an applicant who already holds a PhD degree (even if it is in a different area such as Physics or Math).

I'm sure it's like this at many of the top schools in the US. Here I'm assuming what you refer to as a &quot;doctoral degree&quot; will be treated the same way as the American/English &quot;Ph.D.&quot;