Since nobody else has answered, I'll have a guess. We know the speed of light pretty accurately and have a century or so of experience of propagation of radio waves through various bits of the atmosphere. (A new record, five "of"s in one sentence!)
They probably just ping it repeatedly with radio waves from various ground stations. After a few orbits you'd have the orbital parameters tied down. Once you know the position of one satellite I guess you can use inter-satellite comms to pin things down further. (Assuming you have more than one satellite.)
Millimetre accuracy is a bit of a puzzler, though.