There is I think an application to do this ultraplayer had this feature to do just what you want (see ultraplayer.com) you have to download their player but it did or may still have this feature............................. happy hunting
If you do download this i have some more detailed instructions on how to use this player for recording....... if you repost but please remember copyright laws still cover recording from net radio stations
I used to be a fan of Total Recorder from highfidelity but recently got converted to Audacity from audacity.sourceforge.com which is free and will export files in a number of formats, including wav and MP3. Wav files can then be played on a regular CD player. So I can record something from BBC and listen to it in the car on the CD player there.
rainbow79 - don't know why it may be recording other sounds. Do you have any external sources going into the soundcard? Only what comes through the speaker should be hears, so I wonder do you have a microphone connected up?