550 watts is the recommended minimum.
Barely - maybe. A 6870 will pull about 15 amps. Depending on the CPU, that's 8 - 10 amps. Figure 2 - 3 amps for a hard drive, an optical, and two or three fans.
With a "typical" system you would be running the 12 volt rail at over 90% of capacity. Not a good idea.
Get a larger (500 - 550 watt) PSU.