A few things...
Before the laundry list of recommendations, I should say that the computer you mention sounds like it should do the trick. Just check (as I'm sure you have) to make sure your OS and specs meet (and preferably exceed) the minimum requirements for the sound card and software. That said...
First, and most importantly, two 10-gig drives seems a bit small. Ideally, you would want a 10-20gig drive for applications and a 30-40gig or larger drive for audio. Also, make sure you get 7200RPM drives for maximum performance.
Second, I'm not sure which graphics card you are looking at, but ATI can be a bit tricky with compatibility, at least in my experience. Also, a lot of places recommend getting a dual-monitor compatible video card, as it makes working with many software packages easier. Even if you're starting with one monitor, this may not be a bad up-front investment.
Third, make sure you get a decent power supply. 300 watts or higher seems to be the common recommendation. After that, it's a matter of tweaking components - the quieter, the better. The fan and power supply are two that you can definitely get in "extra-quiet" versions.
Finally, if you can afford it, put another 256MB RAM in. Since memory prices are ridulously low at this point, it shouldn't break the bank. However, if the money can be better put towards drives and components, go that route first.
Good call on the OS, unless Logic and the Audiophile have Windows 2000 drivers, at which point you might want to consider that. I've heard that 98SE is more stable than ME.
In reality, the only two things that definitely bear looking into are the hard drive size and power supply. Upgrade the former, look into the latter.
Best of luck!