A PCI 10/100 NIC is cheap. Should be around 25 Pounds at most I would guess.
How to check: Your network card should have a box-shaped socket in it just like an RJ11 phone jack. (Although I think you guys might use different phone jacks in the UK?)
It should look like this:
http://www.dlink.com/products/adapters/dfe550tx/ This is the type of plug used for 10BaseT UTP cabling, Unshielded Twisted Pair.

This is a standard network card, and what the cable modem company is expecting. Your card may have both this type of port, and an round one with two bumps on it for a BNC connection using "thinnet", or thin coaxial cable. (10Base2) Thinnet is not normally used in modern networking, and would only be found on older network cards.
If you do need to get a new network card, stick with some kind of name-brand PCI card. Intel, 3Com, D-Link, LinkSys... something you recognise. And remember, nothing is "free", if it's a $5 network card, it SUCKS.

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