Is it possible to install a PCi card with a Sata 3 (6Gb) internal connection for my new SSD to achieve my desired 6Gb speed rather than use the mobo Sata 2 connection?
Could you post the model number of your system please?
If the card is going in a PCIe 2 x1 slot (~500 MB/s theoretical bandwidth), then maybe. With the overhead you might get between 340 and 440 MB/s. So, a little bit better than 280 MB/s or so actual with SATA II.
While it is possible its not usually stable.
There are lots of incompatible host bus adapter cards.
Western Digital OEM bundled this card a while back to handle drives larger than 2TB.
There are lots of horror stories about
PEX-SA115 2-Port SATA 6G PCI Express Host Adapter Card -
(Marvell 88SE9128 Chipset)
Not working at all.
You don't say what model, but if it's only SATA-2 on-board, likely just an old Legacy-BIOS system.
On those, you can install a PCIe SATA-3/600 addin card, and it will likely work.
HOWEVER, due to non-UEFI BIOS, it will be hard-to-impossible to get the machine to boot from that drive.
Finally, SATA-3/600 isn't even a fast SSD any more.
The ROCKET 620
Works with Windows Vista, Windows 7, 8, 10, Linux with kernel 2.6.19 and above and FreeBSD