  We present an experimental procedure, based on Meissner effect levitation of
neodymium ferromagnets, as a method of measuring the gravitational interactions
between mg masses. The scheme consists of two superconducting lead traps, with
a magnet levitating in each trap. The levitating magnets behave as harmonic
oscillators, and by carefully driving the motion of one magnet on resonance
with the other, we find that it should be easily possible to measure the
gravitational field produced by a 4~mg sphere, with the gravitational
attraction from masses as small as 30~$\mu$g predicted to be measurable within
realistic a realistic measurement time frame. We apply this acceleration
sensitivity to one concrete example and show the ability of testing models of
modified Newtonian dynamics.
