If I add a SATA 6 Gbit/s PCI Express card to my motherboard, will I get similar speed to native SATA ports built into the motherboard? I ask because I have a SATA 6 Gbit/s SSD and my motherboard supports only SATA 3 Gbit/s.
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?".
SATA3 cards will only benefit from a SATA3 mobo, if SATA3 runs at 6GB/s and SATA2 runs at 3GB/s as you may know. The SSD you have won't be running at it's rated speed due to this limitation, you can either buy a new motherboard that supports SATA3 or stay with SATA2 purchases. So to answer your question don't add a PCI Express SATA3 card if there's an equivalent SATA2 card available, you can save yourself that extra few $ as well.