I might be wrong about this but here goes...
You'd have to cut the barrel off a bit.
Based on the SAAMI diagram, the .308 is 0.4564 at 1.450" in front of the breech face. Doing the math on the .30-06, at 1.450", it is 0.4504, so you'd have a step there of 0.006" and from that point forward the .30-06 chamber is smaller diameter than the .308 chamber, so the reamer wouldn't cut it until whatever point they intersect again. Somewhere on the .308 shoulder, I'd guess.
So, you'd have to cut a bit off the barrel on the breech end and rechamber.
Besides that, you'd never fit a .30-06 cartridge in a short action or a .308 magazine. You could, of course, swap the shortened barrel to a long action.
Of course, a shorter barrel on a .30-06 versus a .308 is exactly the opposite of what you'd want considering the added powder capacity.
Assuming I'm right about the math, that is.