First, some 350W PSUs are fully capable of running a system using an HD5770. So, what we really need is your PSU make/model, or if it's some generic brand, the ratings of it from the sticker on the side.
To make a proper recommendation, it helps to know more about the system itself - Specs, devices, USB devices, etc.
As for Ace Outburst... Never heard of it. Got a link directly to it, or are you expecting us to google it ourselves and do all the work?
All i really want to know is if the manufacturer is reliable.