Hi,
I'm studying Computer Systems Engineering (in the UK) which I assume is similar to Computer Engineering in the US. My course is a combination of the computer science and electrical engineering degree courses. This means that I have covered
C, Java, Lisp, Digital Systems, Linear Systems(analogue electronics), Computer Architechture, Multimedia, Pure and Applied Maths,Statistics, Calculus, Matricices, Discrete Maths(Trees/Groups/Number Theory).
Above all of these courses is the experience I have gained from taking a year out to work as a programmer. I have found that the most useful thing I learnt from Uni was how to program and solve problems. Rather than knowing any specific language very well I was taught lots of languages in shortish stints. Currently I have dabbled with C/C++, Java, Haskell, Z, Perl, Lisp, Fril(like Prolog), Assembler, Machine Code and Html(not a programing language I know) and although I don't claim to be very good at any of them I have found picking up new languages quite easy.
Almost as important as understanding how to pick up a new language is following a disciplined programming style. It is important to stick to the "Golden Rules" of programming ie. Commenting, layout, grouping code into logical sections, striving to make the code human readable even without comments.
On the hardware side I have been taught with Electrical Eng students the basics of digital system design techniques, electronics, digital components(transistors etc). So I have a good idea of the hardware I will be writing software for.
So, what you have to consider is what you are prepared to do for 8hrs a day, every day for the rest of your life. If you want to bash out application software go for CS. If you want to write low level code (down to firmware) go for Computer Systems to get knowledge of hardware. If you want to design + build hardware go for Electrical Engineering.
If I wanted to I could even pick up a soldering Iron an become a hardware guy, but for the moment I prefer software. I like having the choice between hardware and software as I have found great similarities between the design methodologies of both disciplines.
Hope this helps you descide.
Cheers