  We report the experimental implementation of Grover's quantum search
algorithm on a quantum computer with three quantum bits. The computer consists
of molecules of $^{13}$C-labeled CHFBr$_2$, in which the three weakly coupled
spin-1/2 nuclei behave as the bits and are initialized, manipulated, and read
out using magnetic resonance techniques. This quantum computation is made
possible by the introduction of two techniques which significantly reduce the
complexity of the experiment and by the surprising degree of cancellation of
systematic errors which have previously limited the total possible number of
quantum gates.
