Another simple method I have been using successfully for years is simply to launch that cool audio freebie Audacity. In its settings, point it to your sound card as the source, and then hit its Record button when you are ready.
This gives you full control over the recording -- including the ability to easily delete any buffering gaps which occurred during the streaming, increase or decrease the volume, remove unwanted material at the beginning or end, fade in or out etc. Then export it in whatever output format you specify, at whatever bitrate (quality) you want. (MP3s require a small extra free plugin which you can get to through the Audacity site).
This is also the ideal way to get around any DRM-type restrictions on re-recording ANY audio your computer can play through its sound card.
In the unlikely event Audacity cannot record directly off your sound card, a clunky but effective workaround is to bridge your computer's audio output and audio input sockets (generally the little red and green ones which take a standard headphone minijack) with a cable, and then record with that microphone/line-in input as the source.
If anyone is interested in recording streaming audio from the BBC's iPlayer site, there is a brilliant piece of free software which avoids having to go the whole streaming route at all, simply sucking the audio straight off the iPlayer sites in about a 20th of the real-time: it's called get_iPlayer
