First, it's important to mention that in kite aerial photography, which we're assuming is basically equivalent to our balloon photography, it's much preferable to hang the camera rig from the kite line than from the kite (or balloons) itself. This is because the kite line jerks around less than the kite, so the camera stays more stable and you can get a better picture.
There are special hooks for attaching a the rig to a kite line, like these "hangups" (scroll down about halfway), but I went for the don't-order-anything option and tied some non-slip loops in the kite line (EDIT: knots in the line are a weak point, and can lead to line failure. Here's a better solution). Here's a photo with a ruler for scale. The loop on the end of the line is for attaching the balloons.
|The spool of kite line is on the right. Moving right are two loops for attaching the camera rig, about 20 cm apart. The loops were made by tying a bowline-on-a-bight into the kite line.|
|Carabiners (I switched back to those) clipped through the loops in the kite line. The picavet is hanging below, out of shot.|