A reliable method has been developed for making through-bond electrical contacts to molecules. Current-voltage curves are quantized as integer multiples of one fundamental curve, an observation used to identify single-molecule contacts. The resistance of a single octanedithiol molecule was 900 +/- 50 megohms, based on measurements on more than 1000 single molecules. In contrast, nonbonded contacts to octanethiol monolayers were at least four orders of magnitude more resistive, less reproducible, and had a different voltage dependence, demonstrating that the measurement of intrinsic molecular properties requires chemically bonded contacts.