We have studied the adsorption of mercaptopropionic acid, 2,2'-bipyridine, and dopamine onto electrochemically fabricated Cu nanowires. The nanowires are atomically thin with conductance quantized near integer multiples of 2e(2)/h. Upon molecular adsorption, the quantized conductance decreases to a fractional value, due to the scattering of the conduction electrons by the adsorbates. The decrease is as high as 50% for the thinnest nanowires whose conductance is at the lowest quantum step, and smaller for thicker nanowires with conductance at higher quantum steps. The adsorbate-induced conductance changes depend on the binding strengths of the molecules to the nanowires, which are in the order of mercaptopropionic acid, 2,2'-bipyridine, and dopamine, from strongest to weakest. The sensitive dependence of the quantized conductance on molecular adsorption may be used for molecular detection.