Asmedia 1061 PCIe 2x - Benchmark Failed !

Thanks for having compiled the specs in your profile. :) Made this a breeze.

According to your board's specs, your PCI-e x1 slots (the ones you have the current addon card plugged into) are PCI-e 1.0, not 2.0 like the card needs.
The 1.0 has only 250 MB/s of maximum bandwith per lane (pci-e x1 slots have only one lane) for each direction (so 250 read and 250 write). So with overhead and stuff, it's normal that the SSD is capped at around 200 MB/s.

The 2.0 has around 500 MB/s of maximum bandwith per lane (again for each direction), which is going to run a SINGLE SSD perfectly fine.

Given your current setup, I suggest to plug that thing in your second PCI-e x16 slot (the one for the second graphic card), as that is a pci-e 2.0 slot. Will of course work only at PCI-e 2.0 x1 speeds (the maximum speed supported by the card), so it's ok for a single SSD, but not for 2 at the same time. For that I suggest to invest in a card that has a bigger PCI-e 2.0 interface, like x4 or more. This for example can keep happy 4 of such SSDs.
Placing a card in the other x16 slot does not seem to downgrade the other x16 slot (the one you have the graphics card) to x8 as is common with more modern motherboards.

Update them. I need to know what motherboard you have and where did you plug the card into (what motherboard slot).

Also, the card you linked is x1, not x2.

These are the specs of your board. Yes, the board's own SATA ports are SATA II so it's normal that they are worse.

Note the part about "Expansion SLots"
4 x PCIe 2.0 x16 (dual x16 or x16, x8, x8 or quad x8)
1 x PCIe x4
1 x PCI

You see how the x16 slots are of PCIe 2.0 while the x4 slot is not (so is a 1.0)? That means the addon SATA III card works at half capacity (it is supposed to be used in a 2.0 slot not in a 1.0 slot, that has half the bandwith).

Since that motherboard states that can run two full x16 pcie 2.0 slots, and you have a single graphics card, I suggest to try placing that SATA card in one of such x16 slots (possibly the one closer to the graphics card) as that would double its bandwith. Then test again and see if it improves.

Btw, this tutorial allows you to scan and upload automatically your specs into your profile with our forum's own utility program. Please do so, as it is easier to help if we know how it is made.