The Windows version of this tutorial can be found here:
How to Officially Update and Jailbreak to 1.1.3 iPhone Firmware Using Windows
I highly recommend using the Installer method of this tutorial. It is much easier and less prone to error. It can be found here:
How to Officially Update and Jailbreak to 1.1.3 iPhone Firmware Using Installer
To follow these instructions you need to have a 1.1.2 Jailbroken iPhone. You can find instructions on how to do this here:
Windows: How to Activate and Jailbreak Your OTV 1.1.2 iPhone Using Windows
Mac: How to Activate and Jailbreak Your OTB 1.1.2 iPhone
PREPARE THE PHONE
Press to select Settings from your springboard.
Press to select General from the list of categories.
Press to select Auto-Lock from the General menu.
Press to select Never from the list of Auto-Lock times.
Press the Home button to return to your springboard.
Press to select Installer from your springboard.
Press Donate Later to continue into the Installer app. Note: Come back later and donate as this app is helping you out a ton!
Press to select the Install tab at the bottom of the screen.
Press to select the System category.
Press to select BSD Subsystem from the list of packages.
Press to select the Install button at the top right.
Press the Install button to confirm installation.
NOTES***: Make sure if you haven't already that you update to the latest version of Installer. This is very important because the old version won't work in 1.1.3.
INSTALL THE JAILBREAK
Download the Jailbreak from: here
Double click the downloaded file (Jailbreak-1.1.3.tar.gz) to extract it.
Launch Fugu by clicking its icon in the dock.
Enter your iPhone's IP Address into the Connect to: field. Enter root as the Username: and click the Connect button.
If you don't know your iPhone's IP Address you can find it here:
How to Find the IP Address of Your iPhone
If prompted for a password enter alpine or dottie and click the Authenticate button.
Select / from the Remote Folder Dropdown Menu (right pane).
In the left pane navigate to the Jailbreak-1.1.3 folder.
Select all the files in the left pane and drag them to the right pane to copy them to your iPhone
Launch the Terminal application from Applications:Utilities.
SSH into your iPhone by entering the following command into Terminal:
ssh firstname.lastname@example.org (replace the IP I used here with the one for your iPhone)
When prompted for a password enter alpine or dottie.
Enter the following commands to start the install:
The script will download the firmware from Apple once it finishes it will just sit there for a few minutes. Don't worry this is normal.
It will then display some encryption information and sit there for a few minutes. Don't worry this is normal.
The script will then display some partition information and sit there for another few minutes. Don't worry this is normal.
The script will then continue to patch your iPhone. It will take several minutes for this to complete and then it will sit there for a longer several minutes again. Once its complete the SSH connection will terminate!
After about a minute your iPhone will reboot into a Jailbroken 1.1.3 iPhone !!!!
NOTES***: You can find a solution to the iTunes sync problems with this jailbreak here:
How to Fix Sync Problems After Official 1.1.3 Soft Jailbreak
THANKS***: asap18, bgm, Bugout, bushing, chris_, davidc__, dinopio, drudge, Fred_, ghost_000, gray, MuscleNerd, NerveGas, netkas, np101137, planetbeing, pr3d4k4t, pr3d4t0r, pumpkin, roxfan, Turbo, Whiterat, Zibri, Zf