  We report the results of new differential force measurements between a test
mass and rotating source masses of gold and silicon to search for forces beyond
Newtonian gravity at short separations. The technique employed subtracts the
otherwise dominant Casimir force at the outset and, when combined with a
lock-in amplification technique, leads to a significant improvement (up to a
factor $10^{3}$) over existing limits on the strength (relative to gravity) of
a putative force in the 40--8000 nm interaction range.
