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This modem is directly plugged into the PC (it's an old ISDN DSL modem) and relies on the XP dial-up to work. It's not a router by itself. I imagine this would be a problem.

I'm guessing it would still work the same. Both the cable router and the DSL modem need an upstream gateway set somewhere. Even if they get that gateway addres via DHCP, it's still a valid upstream router. I'll bet you could still do it.

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what software could I use to do this? I'm not afraid to tinker a bit with it.

There is a whole class of utilities for Windows called "Personal Proxies", some of them freeware. Mostly only used in special situations, and I think this is specifically one of the situations they're supposed to be able to handle. It's been a while since I messed with one, and I don't remember any names. Perhaps someone else can recommend one?
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Tony Fabris