Bestec isn't a known brand to me, but the 4670 is actually pretty efficient running off just the PCI slot. The official recommended PSU is 400w, which is usually an inflated figure to compensate for low quality PSUs. Whether or not your PSU is too low of quality is really the deciding factor.
Here is a chart showing the consumption of a 4670 equipped system (with a C2Quad 9450) at 67 watts idle, 139 watts under load playing Crysis.http://www.anandtech.com/video/showdoc.aspx?i=3405&p=9
Personally, if it was my rig, I would be prepared to replace the PSU, but would try it anyway. I would have to do this knowing the risk of a PSU failing and taking system components with it.