Well crap, it is not multi rail at all and by today's standards in only about a 250(if even that) watt power supply.
It is an older design with more focus on 3.3 and 5 volt power. Those rails are far less used these days. And modern power supplies deliver 80 to 90% or there total power on the form 12 volts and they can use part of that to generate there 5 and 3.3
A 6770 would be fairly close to the limit on that power supply. Too close for comfort for many users. Unless you could ensure the rest of the system was consuming as little power as possible it is hard to say it would be safe.
My i4 750 has been under volted to take even less power and runs with a 5770 on a 300 watt power supply(but it has 22 amps to work with).
My system has not personally reached the limits of even your power supply, but it is not the same as yours and i can not guarantee you will not blow that power supply up.
Here is a chart with some video cards and there power use. a 6770 and 6750 take about the same power as there 5770 5750 parts. As you can see there is a good chance of maxing out the power supply under very high usage with the 6770(5770) once you add the cpu and other components into the mix.
http://www.techpowerup.com/reviews/AMD/HD_6670/20.html