Hi All! I was shocked with the amount of people complaining about this error with no solutions. I have solved mine the rather hard way. If you are interested I'm reposting 3 posts I made over here:
http://www.camerahacker.com/Forums/DisplayComments.php?file=Digital%20Camera/Sony/Sony_DSC-P41_C.13.01_main_circuit_board_problem.html
The next 3 posts will be the ones from the other forum.. I had to break it up to get it to post.
-----1ST POST-----
WOW! I can't believe all of these differen't camera models coming up with the same error code and Sony doesn't know what it means. Fock Sony!
Ok, obviously I am here because my DSC-P8 3.2M Cybershot is doing the same thing, "C:13:01 memory stick error".
Here's my chain of events:
I was previewing the last picture I took using the left arrow, and noticed it was way too dark, so I deleted it. Right when I selected OK, the error message C:13:01 was flashing on the screen. There were pictures on the card still, and with my built-in card reader on my PC I was able to transfer them over and they all look fine.
Here's a list of all of the things I've tried and the camera still displays the C:13:01 error:
1) ejecting and re-inserting the 256MB Pro card.
2) Locking/Unlocking the 256MB card.
3) Tried an old 16MB card and it does the same thing. I know this card is good too, so I suspect a hardware failure with the camera. Most likely the interface to the memory card.
4) Formatted the 16MB card with the PC.
5) Formatted the 16MB card with Sony's Memory Card Formatter.
6) Tried putting my camera in various USB connect modes with the cable, i.e. "normal mode" and "ptp mode".
7) Cannot format the card in the camera. When I try it just displays the error, further proof the memory card interface is the problem.
Anything people suggested above that could possibly help didn't - I think I covered them all in my list here.
I sort of wish I had purchased the 4-year warranty at Best Buy now, but then again...I don't feel that I ever needed it. I have treated my camera very well and it has served me well for over 3 years, 5500 shots and many videos. I've never had to replace the battery and the charge still lasts for over an hour. The extra warranty was $60 on top of a $400 camera with a $100 memory card, but that would be $60 now that I can spend towards a better camera. I've been wanting to get a new digital SLR with external flash. The smaller point-and-shoot cameras are nice, but I've always battled with the dim lighting. If it's not sunny, or very well lit, it's almost impossible to take a shot that doesn't require some photoshoping later. Not to mention the graininess is hard to fix. I realize I may still suffer from this same situation with a more expensive camera, but the answer is out there somewhere.
I hope Sony can own up to this error code and everyone here can get their cameras fixed, but I'm not going to hold my breath on that one.
Canon and Nikon do look very tempting 
-----2ND POST-----
HOORAY!!! IT'S FIXED!!
So after doing everything I said I did in the above post, I decided to crack open the case and have a look at the connectors.
WARNING: Do not attempt this unless you have a very small set of philips and flat head screwdrivers, 5x jewelers loupe, fine needle nose pliers, a very small set of tweezers, a working knowledge of electronics and FPC (flat printed circuit) connectors and can accept the fact that you may actually break your camera more than it is already.
Remove the battery and memory card.
I have a DSC-P8, so there are just 4 screws holding on the front cover (the cover around the lens). Remove those and pry with your fingernail in the crack between the memory card door cavity and the front cover to pop it loose. Carefully remove it and follow the instructions in the following picture (several links in case some go down):
http://img221.imageshack.us/img221/3709/sonydscp8c1301error4wh.jpg
http://rapidshare.de/files/23467590/Sony_DSC-P8_C1301_ERROR.jpg.html
http://www.everybodytothelimit.com/sony_dsc-p8_c1301_error.jpg
I hope those of you with other models can "figure it out", because from what I can tell the service manuals are not free or cheap.
Email me if you wanna chat about this:
emailbrett [a-t] gmail [d-o-t] com
-----3RD POST-----
Hmm... just remembered I took that above picture in the mirror and everything is going to look a tad backwards if you open up your DSC-P8. Attached is a mirror image for reference.
http://www.camerahacker.com/Forums/Digital%20Camera/Sony/Sony_DSC-P41_C.13.01_main_circuit_board_problem.files.hidden/Sony%20DSC-P8%20C1301%20ERROR%20mirror.jpg
http://img102.imageshack.us/img102/7272/sonydscp8c1301errormirror0kf.jpg