According to eMachines' site, the EL1200-05w has:
CPU: Athlon64 1.6GHz
GPU: GeForce 6150SE
RAM: 1GB DDR2
I'm pretty sure that if the motherboard uses a GeForce 6150SE integrated chipset that it would have a PCI Express slot, the 6150SE is part of nVidia's nForce 430 chipset. The nForce 430 does support the PCI Express (PCIe) interface and thus should have a PCI Express slot available for a better graphics card.
However, a GeForce 5 series is not as good as the GeForce 6150SE at least in terms of compatibility (the 6150 is a GeForce 6 series chip). If you do have a PCI Express slot, you are best off buying a new PCIe supported graphics card. It is true that a single core CPU will bottleneck a fast graphics card, so you shouldn't look too high. I would think you could get a considerable performance boost with a sub-$100 graphics card.
Looking at the under-$50 range, you can get a GeForce 9500GT which is considerably newer than your 6150SE and supports DirectX 10. It should be plenty for World of Warcraft and anything based around the Source engine (I play Team Fortress 2 on my 8600M GS laptop frequently, the 9500GT should be better or at least very close to the 8600M GS).
Newegg.com - SPARKLE SFPX95GT1024U2H GeForce 9500 GT 1GB 128-bit GDDR2 PCI Express 2.0 x16 HDCP Ready Video Card
There are some better cards if you go for the under $100 range, but you'll probably run into bottleneck issues above that.