Thanks for your recommendation. The CTA100 looks like just the thing.
Unfortunately, it seems to present problems of its own. For starters, it appears to be discontinued? I found one for sale on eBay but it has no software or documentation. And they use RS-232 ports, which most PCs don't have anymore. Mine doesn't and I don't want to add a kludgy USB/RS-232 converter to my laptop -- already went down that route for my PDA and it was a pain to find one that works well.
So, I'm going to try experimenting with having the modem dial and embed commands in the dial string to transfer the call after dialling. No problem with dialling; just use:
ATDT9,<telephone number> ---> means Attention, Dial, Tone, get line, pause
The modem will stay on the line at that point. It gets harder now. I've tried adding:
!*70,21<directory number>H0 ---> means Stay in Command Mode, Flash hookswitch (Link), transfer call, pause, hang up
This part doesn't work. I'll keep experimenting and post the results if I can get it working.
Thanks everyone for your help, and best regards, Greg.