SATA II nor III does not exist. That naming has been abandoned by SATA-IO long ago. Calling it SATA II or III can just mean SATA.
The correct way when speaking about SATA rev. 3.0 is "SATA 6Gb/s" (note the small case 'b' which indicates bit and not byte).
Anyone who tells you SATA II or III is speaking bollocks.
As all above posters have said, SATA 6Gb/s requires a new SATA controller, either as integrated into chipset/southbrige of motherboard or as expansion card.