Man, rooting just keeps getting easier and easier every day for new devices! Thanks to TGA_Gunnman over at the XDA forums, this One-Click Root process will copy over the rooted-update.zip file then reboot your Samsung Captivate. Follow the instructions in the CMD prompt that launches.
Download the app here.
Requirements: Dot Net Framework version 4
1. Make sure your phone is on USB Debug mode: (MENU > Settings > Applications > Developement > USB debugging = Checked )
2. For Windows x86 and x64 systems make sure that the Samsung Drivers are installed. You can get them here.
3. Extract the contents of GalaxyS_Captivate_One-Click_Root.rar to a folder.
4. Connect your phone to your PC via the USB cable and launch the file "AT&T Captivate One-Click Root.exe" file.
5. Click the "One-Click Root" button. This will launch a command prompt follow the instructions in the CMD window.
Done! You are rooted.
To UnRoot follow the same instructions except click on "One-Click UnRoot".
For full instructions and to download the One Click application, check out the application thread
(Big ups to XDA!)