Playing CDs in computers' CD drives

For some reason, which answer I suspect but seek confirmation for, disc 1 of my copy of Genesis’ “The Lamb” won’t even read in my computer’s CDR drive, but disc 2 will. On examination of the discs, I have noticed that disc 1 has a transparent rim, that is to say that the outer edge of the CD is see-through. This is not the case with disc 2. I figure this has something to do with it.

This is particularly annoying, since I typically compress and save all CDs I buy to --aps mp3 files and listen to them primarily on my PC. (Since I own the store-bought discs I assume I have the right to do that, so this is an OK question on this board, right?) I caught a snippet of conversation on the 'net that leads me to believe this problem may be rectifiable, but I may have caught it out of context and the proposed solution is not something I want to try unless I hear it supported by others. (IIRC, the suggestion was to take a black sharpie and fill in the transparent rim, though I don’t know precisely the method.)

Has anyone encountered this sort of problem, and how do you suggest I solve it?

take it back to the shop and get a replacement ?

Will disc 1 play in your stereo? If so, you can hook up a patch cord from the line out jack of your stereo to the line in jack on your computer and use a recording program like Audacity, sox or messer to record the audio stream. The downside of this method is that recording will take longer than a CDDA rip through your CD-ROM drive.