This is a quick summary of how to flash your Samsung Galaxy S III i9305 with a custom ROM.
Okay, the title is not right because I didn't use the only free software only Android distribution Replicant but Cyanogenmod (CM) instead.
Before I got my new phone I read on the internet that it's almost identical with the very popular and well supported Samsung Galaxy S III i9300, but has 2GB of RAM instead of 1GB and additionally supports LTE. Don't be fooled, this means it has a different chipset and needs different drivers. But Samsung isn't very helpful with supporting the necessary information to get those drivers into ROMs and you're probably better off with a Nexus device.
This leads to the problem that there is no officially supported CM-ROM for this nice phone. Too bad. But luckily there is an unofficial ROM. Plus a ton of pitfalls to brick your phone.
Let's get started. I did this all on an up-to-date Ubuntu 12.10. I went to the official CM page for the i9300 thinking it would work on my i9305. IT WON'T WORK, IT WILL BREAK, DON'T TRY IT! The general instructions will still work, but you need different ROM images.
The basic idea is to install a recovery ROM which essentially is a very small rescue system able to flash other ROMs and use this to install CM.
heimdall is a command line utility to flash files on your phone. It's the open source alternative to Odin and can flash the same files. Just rename the Odin files ending in .tar.md5 to .tar, untar them and use the extracted binary image file (*.img) with heimdall.
Download heimdall, but use the 32 bit version. I couldn't get the 64 bit version to work at all on my 64 bit system.
Do not connect your phone to the USB. Hold down Volume Down, Home and Power in order to boot into the built in flash tool. You should see a screen telling you that you'll void your warranty and so on. Press Volume Up to confirm that you know what you're doing.
Now connect your phone to your computer and open a terminal. Type
where recovery.img is the unpacked CWR ROM. Your phone will be flashed within a few seconds.
Now disconnect your phone and choose the option wipe data/factory reset. If you don't do this your phone will not boot.
Turn off your phone, get the unofficial CM 10.1 image found on the xda-developers announcement forum thread. (It's also a great resource to read). The trick now is to get the .zip file on your phone. You can't use adb to copy the file on the phone over the USB at this moment because this requires a running Android. And you rather don't want to anyway, but instead use a SD card. Just put a micro SD card into an adapter and use it on your computer (or use another phone with a rooted Android and SD card slot to download the file onto it). Don't even think about using your stock Samsung firmware for that. You need root and developer extensions enabled.
You might also want to download the closed source Google Apps for CM 10.1.
After you placed your CM image and the Google Apps on the root of your SD card and put it into your phones SD card slot, hold down Volume UP (not DOWN), Home and Power to boot into CWR. Then choose to install a ROM from external SD card and install first CM and afterwards Google Apps. Now you should be ready to reboot and enjoy your rooted, more free than before Samsung phone.