Thank you all for taking so much time over this. Here's the URL for the deck I'm trying to use so you can see it for yourselves.
It's got a fixed lead with two phono plugs, and a removable USB lead, which is the one I'm trying to work with.
It can also record directly to CD - from vinyl or tape - but I haven't tried that yet because I'd rather not have to use (and waste) an intermediate device, especially as so many CDs would be needed.
It's covered in buttons (16 just on the front panel), and so is the remote control, but I can't see a gain control anywhere. It does have two buttons (+ and -) marked "recording level", which should be the same thing, but I don't think they have any effect on USB output. I'll try them again though.
One thing that makes everything that much more difficult is that it doesn't have any audio output of it's own (no speakers), so I have to start recording in Audacity to hear anything.
The computer was purpose-built about 3 years ago (£2,000), and I'm running Ubuntu 12.04 with Audacity downloaded via the software centre (center), which tells me it's version 2.0.0-1 ubuntu 0.1.
I'll have a play tomorrow, and I'll let you know what happens, but I'm expecting visitors, so may not get much time.