  Precise measurements of tiny forces and displacements play an important role
in science and technology. The precision of recent experiments, while beginning
to reach the limits imposed by quantum mechanics, is necessarily spoiled by the
unavoidable influence of noise. Here we obtain a quantum limit for the
uncertainty in the estimation of a resonant classical force acting on a noisy
quantum-mechanical oscillator. We determine the best initial state of the
oscillator and the best measurement procedure, thus getting a rigorous and
useful benchmark for experiments aiming to detect extremely small forces.
