Lars,
(by the way, you've got a classic Swedish first name), if your gun can accommodate Versa-Pod's bipods you can go fairly low. My gun has what I believe is their prone position AR15 type bipod:
It expands, I think, to some 9" so you normally need a table, a log, or something in the terrain to rest it on. I must admit that I've mainly used it for testing ammo at the range. Would recommend you to take a look at the Versa-Pod products anyway, they do make good stuff.