I personally have a gts250 (which is like a 150 watt part) in a DC5800 microtower. It runs great. I am using a double molex to pci-e power wire. The computer never draws more than 250 watts from wall acording to my kil-a-watt.
The 7770 is only a 80 watt part. It will be fine. That core 2 duo with the 7770 will never draw more than 200-225 watts.
I recomend every one get a Kil-a-watt or somthing similar. I use mine all the time to see how much energy my toaster uses and stuff like that. The thing cost like $15 and its fun. You could load up a stress test like prime, or even folding@home and with the kil-a-watt see how much that PC can draw. I bet the draw will be 100-120. At that point you know you got 180 to 200 watts to work with. If the 7770 draws 80 watts your in the clear.
Also keep in mind that depending on the effeciency rating of the PSU a 300 watt unit may draw 10-20% more from the wall. So lets say your PSU is 90% effecient it can safely pull 330 from the wall, with 10% being lost as heat, and 90% or 300watts powering the PC.