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There's no modem needed, actually. All of that can be done in software, and you can configure a desktop as a PPPoE client (that's the protocol your router uses to log into your ISP's network and receive internet connectivity). Obviously, you'd need to configure that PC as a router for other computers to also share the connection, and running a typical interactive system 24/7 as a router is hideously inefficient in terms of power use.
Some ISPs just hand out an IP with a DHCP server directly on the line, so it might be even simpler.
Setting up your own PPPoE client is extremely easy, so either way you most definitely can do what OP wishes for.