I used a terrific program called Audacity which is a free professional-quality audio editor that's available for both Windows and Mac OS X. I've actually written about Audacity before (see professional audio editing results with Audacity, for example), but what I did for this audio edit was a bit more advanced.

To start out, I downloaded and imported the mp3 audio file of the original radio broadcast, all 55 minutes and 03 seconds of the original broadcast with bumpers and the choppy end segment. Listening to the audio while watching the time marker, I ascertained that it was approximately 1:23 into the broadcast (notation: 1:23 is "one minute, 23 seconds") that Rob switched from show introduction to the beginning of his interview with Brian.

You can hear that in this four second snippet: original 1:22-1:26.

In Audacity, it looks like this:

Notice the two magnifying glass icons on the top, too: with the "+" it zooms in, letting you get a high level of granularity with your editing work, and with the "-" it steps back, letting you see more of the waveform in the same space. Those are critical tools for any sort of work!

Anyway, here I have isolated the few seconds of audio that are Rob just beginning to introduce Brian on the show and highlighted them. As usual, I again press "play" to ensure that this is what I think it is, start to end.

It is, so the next step is to choose Effects --> Fade Out, which then, as you can see visually, gradually lowers the volume of the audio segment down to zero:

Still not quite enough, though, as I want it to go to zero really fast, so I'll again apply the Fade Out filter. Now it looks like this:

The next step was to ascertain the end of the passage I wanted to delete -- in this case, Brian's entire time behind the mic (sorry Brian!). That occurred at approximately 27:30 followed by an audio bumper (music that's used as a filler or to segue between adverts and content) and Rob comes back into the program at 28:03. So I made a selection from the very end of my faded out audio portion (1:24) to just before the beginning of the audio bumper (28:00). It looks like this:

At this point it's time for The Big Cut. A quick Cmd-K (or Edit --> Delete Selection) and we just chopped 27 minutes out of the 55 minute audio program!

Now I can see the fade out and the rather sudden introduction of the music in the new passage:

I'll move the selection just a bit, to catch the first few seconds of the music:

and with that selected, choose Effects --> Fade In, which adjusts the waveform to be zero volume ramping up to full volume:

Finally, I'll trip some of the blank audio to tighten up the transition, and it's done:

You can now hear that same four seconds in the recording you heard earlier, but now the fade out --> fade in occurs in the middle of it instead: edited 1:22-1:26.

That's how I did what you're asking about. Elapsed time? 10-15 minutes or so. I could probably have been a lot faster if I wasn't busy taking screenshots too! :-)

Finally, you can hear the full edited audio here and the entire original broadcast here (look for the show that aired on 11-24-07).