You can register it but you can't do it for free. You normally do this through a 'registrar', which is a firm which can offer you additional services (such as e-mail forwarding for your address) if you need them.
Most UK-based registrars will register domain names through Nominet (who control .co.uk registrations). You can find a list of all Nominet registrars here:
http://www.nominet.org.uk/governance/members/l ist/
However, to find firms which specialise in
domain name registration, it's probably easiest to search for that phrase on Google.
Prices vary depending on your chosen extension. e.g. 123-reg charge �2.59 per year for .co.uk but �8.99 per year for .com. Some addresses (e.g. .uk.com can cost much more).
http://www.123-reg.co.uk/domain-names/your-int ernet-domain-name.shtml
Chris