Thanks for your help Dark Mantis. I guess you're right about the need for a separate power source. (I'll be able to test it when I get the board!)
Thinking about it, it seems pointless having a native eSATA port that's unpowered - better to install Gigabyte's PCI eSATA bracket, since at least then you can connect it internally to the PSU, whereas an unpowered eSATA port doesn't have that facility.
Thanks again.