Downgrading from iOS 7 to a recent full backup of iOS 6.1.3 I have a device (iPhone 4S) that is running iOS 7, but I was wondering: Is it possible to restore a full backup of the same device running iOS 6.1.3.

I'm not worried about losing data from the iOS 7, but I would like to restore the device to its previous state under iOS 6.

Is this possible?

Again, to reiterate: I'm not looking for clean downgrade/install of iOS 6, I wish to restore a full back-up containing an earlier iOS. Thanks.
Note: this will only work if downgrading from a prerelease beta OS to a regular version.

You can only ever restore an iOS device to the latest public version.

Downgrading is super easy: Just put your iPhone into DFU mode (plug into iTunes, power off, hold down Sleep/Wake and Home button for 8 seconds, let go of the power button and continue to hold down the home button until iTunes alerts you. The screen should be black) and follow iTunes through there.