Like mentionned here, the 7600GS/GT all stay within the 60W power envelope (the 7600GS consumes under 50W even under full load), so neither of these cards need a PCI-E power connector. Your stock Dell PSU should be able to handle either of these cards.
As far as the x850 series (similar performance to the 7600GT, but draw more power), and the 7900GS, I can't tell you for certain that your PSU will cut it. THe 7900GS is based off the 7900GT, which can draw 80W or more under load (I've seen a few measured benchmarks of this, but I don't have any links at the ready). These certainly require an extra power connector, and might draw enough current to overload your DELL PSU (likely 300W). Just something to watch out for.
Happy New Year !!\
edit : Dell/Gateway/HP, I didn't notice your brand wasn't DELL but they're all comparable when it comes to low power PSUs.
Also, since a 7600GT can routinely be found for 100$ or less (someone on these forums spotted one for under 80$ recently), and the 7600GT is no slouch by all means (especially compared to whatever embedded/x300 horror that came with your PC), it would be the best solution (in my opinion) for a cheap but worthwhile upgrade. THe 7900GS and x1900GT/x1950Pro PCI-E can be found around the 150$ range nowadays, if you want an even more powerful card - but then you will likely run into stability problems (insufficient current from your PSU).