two things to keep in mind:
1) with a CS degree, you will be able to find a job as a programmer in ANY industry. Given how scarce jobs in the games industry are and how many are trying to get into that industry, that is a big boon should you either not be able to find a job as a game programmer, or grow tired of the industry (the rate of people switching to other industries seems to be rather high too).
Granted, the longer you work, the less worth your degree is compared to your work expierience. Given you work in the games industry for a long time, your choice of degree will no longer matter if you try to switch industries. That choice between CS or Game Programming as a degree pales compared to 10+ years of expierience as a game programmer. You will most probably not be as highly rated as a specialist in the industry you switch to, but given demand for programmers is high enough in that industry, you will land a job (given you don't suck at interviews and the whiteboard tests). You are a senior programmer, and while most probably not specialized in the skills the employer is looking for, should be able to quickly adapt to the new role.
2) You will most probably have just as much of a chance finding a job as a programmer in the games industry given you can show related personal projects. Which you should do anyway, as no school in the world a) trains you nearly enough to be a competent programmer, it will just teach you the basics, b) will give enough "realworld projects" to really build a portfolio of work you can show (and given how competitive the games industry seems to be, you really should have a portfolio ready if you look for a job).
If you do a CS study, you will need to learn the games specific parts in your freetime and try to build some games and showcases, both to deepend your knowledge and have something to show.
If you do a Game Programming study, you will need to deepen the games specific parts in your freetime and try to build some games and showcases.
In the end, both studies will give you almost the same: just the basics. A game programming course might give you a little bit more basics specific to game programming, but you shouldn't expect college/university to train you indepth in game programming. To really become competent, you have to sink a considerable time into working indepth in a certain field. Colleges/universities always try to cover a broad range of basics, with never enough time to really get indepth.
If you really want to become attractive for future employers, you will need way more than just that basics and the degree.
Thus the choice of your degree shouldn't matter too much for what you have learned in practical skills at the time you finish your studies. Most of the skills you learned will not be learned during school hours.
The question is if you want the added job security of a CS degree or if you are ready to put everything on one card, and as Sean said above, if you are interested in CS enough to study it without a strong game related focus.