Adding a SATA 6 Gbit/s PCI Express controller card?

It depends on whether the PCIe SATA adapter uses the 1x or 4x bandwidth capability. The PCIe 2.0 specification defines 1x slot bandwidth as 500 MB/s, which is much higher than SATA-II. If you were to purchase such a card (like this one), it has a data transfer rate of 4.69 Gbps.

SATA uses 8b/10b encoding, which translates to a raw maximum speed of 469 MB/s. Compared to SATA-II, which caps out at 300 MB/s, you have 1.5x the bandwidth. Not full SATA-III, but a lot better than your motherboard's native SATA-II ports.

Note that this is limited just by the PCIe 1x port. Using a PCIe SATA adapter that can utilize a 4x slot instead, you can remove this limitation and are now at full SATA-III speed. (And such adapters do exist.)

Finally, note that if you have an older motherboard (i.e. PCIe version 1.0), your bandwidth will be exactly half of what I've listed above.

Whether or not you actually need the increased bandwidth is another issue altogether, and I would recommend that you see my answer to "Is it worth to get a SATA 3 controller to max the [drive] out?".