What brand and model is that "300W" PSU? If it's a Bestec, it's junk and should be replaced.
I think Fantastik typo'd the GPU. It is the HD6570 that uses only around 58W; the HD6770 has a single PCIE power connector on it. Since a PCIE slot can provide up to 75W, that means it uses more than that (I believe it's around 91W; http://forums.atomicmpc.com.au/index.php?showtopic=264
. The HD6750 also has a PCIE power connector. The HD6570 does not. The strongest card you can get that doesn't need auxiliary power is the HD7750. If that card is not available or is too expensive, a HD6670 would be your next best choice. Make sure you get a version with GDDR5 memory on it, not DDR3. Honestly, any of these are vastly superior to your integrated 4500 Graphics Media Decelerator.
Edit: A 300W PSU will typically not have a PCIE power cable on it, since it is too weak to power most graphics cards that need one.