300 watts is enough to power the card and the computer. Unfortunatly as others have said that most all 300 watt psus, including the ones in OEM computers, are not very good quality. They are not made with quality parts for longevity, and if they are really cheap then that 300watts is a mystical number that it will never achieve until 2 seconds before it dies and takes all the other parts with it.
PSUs also degrade over time, lookup capacitor aging, and thus even a quality psu will not be able to power a system at max forever and in a few months to a year wont be able to do it anymore.
The best budget PSU in the range you need right now is the Antec Basiq bp430. FYI the corsair cx430 has had many issues with it not lasting more then a...