If he's not changing anything else in the system then, yes, that's probably enough to give him the same performance he's getting now. I don't think the 770 chipset sucks down any more power than earlier chipsets, though there might be some extra power draw from other on-board components - you'll have to watch for that. If he's using standard audio now and starts using HD audio on the new board, that could be a problem. If he starts using more USB slots or adds another HDD, that could be a problem. Even adding a fan could be enough of a draw to make a difference. 300 watts is pretty low for any modern system but with few extras and no OC it might be enough.
That being said, everyone else's comments are very pertinent. Power supply quality is more important than the published wattage. Other components and their use will also dictate how much power is required. In my opinion (and I run everything from s939 through an AM2+ 940BE) a 300W PSU is really pushing the lower limit ...