Sata 3 (6Gb) connection for SSD desired

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.

https://www.amazon.com/High-Point-Bananas-Leather-9042351/dp/B002VEWBGO/

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.

https://www.youtube.com/watch?v=yAAyEnHvlq0

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

is

• SATA 6Gb/s = 600MB/s
• PCI-Express 2.0 x1
• Plug-n-Play: no software drivers needed
• Includes legacy boot rom and understands 48 bit LBA aka larger than 2TB drives.

Works with Windows Vista, Windows 7, 8, 10, Linux with kernel 2.6.19 and above and FreeBSD