And a 6 G (SATA II) cable is the same as a 3 G cable (SATA II).
Who told you that the 3 Gb speed speed (be careful, a GB is eight (or sometimes 10) Gb) is due to the cable? Could that person be wrong?
No HDD can transfer at speeds over SATA II speed, anyway. So even if the mobo, the cable, and the HDD controller are SATA III, you won't get SATA III speeds. With an SSD, maybe, but not with an HDD. Ever.
Info on make and model of notebook and drive may help us be more specific.