How can I copy my PS2 saves to my PC?

I'm considering crating up my PlayStation 2 in order to free up inputs on my TV. I don't play it very often any more, and it looks like emulation is currently good enough for the games I still play on occasion. However, I definitely don't want to start over in all of them!

As the PS2 has USB ports, it seems likely there's some way to copy games from a memory card to a USB drive (using a custom disk, presumably). However, Googling for it gives me a lot of contradictory (and fairly old) information about what software is available and how well it works (or for that matter, whether it's even necessary).

Can someone who's done it offer advice on transferring my existing saves from their memory cards onto my Windows 7 PC? If at all possible, I need a software-only solution — my PS2 is unmodded and I'd prefer to leave it that way. I envision downloading a legal(!) ISO which will allow me to perform the memory-card-to-USB transfer on the PS2 and which produces save files which can be used with PCSX2.

This can be done with the uLaunchELF browser. uLaunchELF is homebrew software, so must be run on a modded PS2 or via other methods for running unsigned software (the same techniques can be used to run pirated games, so I won't go into detail).

Once in the browser, copy your saves from mc0:/ or mc1:/ (the first and second memory card slots) and "psuPaste" them to mass:/. This will produce files which can be used with mymc to produce PCSX2-compatible memory card images.

If you have access to a PS3 and a PS2-memcard-to-USB converter, you can copy the PS2 saves to the PS3. Then, from the PS3 you can copy the saves to a USB drive which should be accessible from a PC.

I can't say if the resulting files are compatible with PCSX2 though.

Second @EmSixTeen's answer re: uLaunchELF. You say you don't want to mod ... I don't think you'll be able to do this without some kind of modding. You might look into FreeMCB and OPL, they are handy 'softmodding' tools that install a mod on your memory card - so, no dealing with mod chips, at least. You'll probably need something like a GameShark / CodeBreaker to install it though - and I believe you can just use that (GameShark) to copy your saves.

(If you decide not to box up the PS2, the homebrew stuff is pretty handy - yes, even for non-piratey uses.)