I'm running one of these on a board with SATA-II. I get 270MB/s real-world read speeds, due to some overhead. With SATA-III you'd expect to get whatever the benchmarks are, likely around 540MB/s. The thing is, though, that this really only affects transfer speed. Random access time is still comparable, it's just that when it goes to read it takes twice as long on SATA-II, or phrased differently it would be half as long on SATA-III.
But the question you need to ask is three-fold. One, how long does it currently take to read. If it's a 500ms action then SATA-III can save you up to 250ms, which is not noticeable. On the other hand, if reading is the main delay and causes a 10s wait time then SATA-III could save you 5s. That's worth considering.
Second is if you could read at twice the speed, would the rest of the system be able to keep up? Maybe your CPU can process 350MB/s of data, in whatever it is you're trying to do. In that case SATA-III wouldn't bump you up to 540MB/s, it would bump you to 350MB/s. Now we're talking about a 30% transfer rate improvement, or roughly, a 10s action now taking 8s. Not so worth it.
Third is whether you added in a SATA-III card it would cause any other effects. For example, maybe your board has PCIe 2.0 slots. You need your PCIe card to have at least as much bandwidth as SATA-III, which is certainly the case if you only attach 1 SSD. But with, eg, 4 SSDs a SATA-III card needs to use more lanes in order to be able to provide 6Gbps to 4 devices. You'd need 24Gbps there, so it may make sense to try and get a 2 port SATA-III PCIe card where you'd only need to provide it 12Gbps. A PCIe 3.0 lane is 7.88Gbps, while a PCIe 2.0 lane is 4Gbps. So you'd actually need a PCIe 2.0 x4 card, or if you just attach the 1 SSD then a PCIe 3.0 x1 card would work but a PCIe 2.0 x1 card would not be any improvement. Many motherboards which don't have SATA-III also don't have PCIe 3.0. So odds are you're looking at a PCIe x4 card in order to get this.
Related to that, if you insert a PCIe x4 card in your motherboard odds are it will cause your GPU to run with an x8 link. And if it is true that your board doesn't have PCIe 3.0 then your new GPU will be running PCIe 2.0 x8, a quarter of the bandwidth that it's capable of. Now in the real world, IIRC, that's about a 5% loss in performance. I haven't seen the scaling for the 3xx series, but they're very similar to the 2xx series so I'd expect that to translate. If you do have PCIe 3.0 lanes, then a PCIe 3.0 x1 card should provide enough bandwidth for a single SATA-III SSD. However it's worth looking into the block diagram for your motherboard as most PCIe 3.0 x1 lanes interface through the motherboard whereas most SATA ports on the motherboard are connected to the CPU. This means that accessing an SSD through such a PCIe slot is an extra chip away from the CPU, which may well impede the transfer speeds and/or access times.
As a completely other alternative, if you had two SSDs you could put them in RAID0 and, assuming neither one ever dies, you should have 540MB/s or so read speeds.