The ads are often longer than the morsel of media you get, and I watch these 2-3 constantly repeated ads enough on BBC World to not feel guilty about not seeing them on news.bbc.co.uk.
Step 1: Install Firefox and the Greasemonkey Firefox extension.
Step 2: Install RealPlayer. This would work just as well using Windows Media Player instead of RealPlayer, but RealPlayer is my (fairly arbitrary) preference.
Step 3: Watch this YouTube video to see what to do, or see the full resolution video here:
Here's the code required: bbcadremover.user.js