what server-side tools do you have with your website? asp, php?
one way of doing this is using their RSS feeds, like this one:
http://newsrss.bbc.co.uk/rss/newsonline_uk_edition/front _page/rss.xml
basically, an RSS feed (ignore the RSS bit), is just something that gives you the content from the BBC, without any of the other stuff that goes into their site. You're free to grab their feed, and put it into your site. Then you can style it too, so it fits into your design.
lots and lots of websites have feeds now (not all RSS, lots use Atoms feeds, which are better).
anyway, to insert this stuff into your site, you need something server-side (with asp, php or some server-side technology like that) to grab the RSS feed, and turn it into some nice HTML for you to use. an example of this is called magpierss:
http://magpierss.sourceforge.net/
this is free to use, written in php.