A mechanical experiment has been developed that measures an upper bound for the viscosity of a lipid bilayer membrane. In this experiment, strands of membrane (tethers) are formed from phospholipid vesicles attached to micropipettes by subjecting the vesicles to fluid drag. The rate of tether formation is measured as a function of the velocity of the suspending fluid. The surface viscosity can be calculated from this data using a theoretical relationship derived in a companion paper. Because of the multilamellar character of the vesicles, these values provide an upper bound for the viscosity of a single bilayer. The smallest values obtained in these measurements fell in the range 5.0-13.0 x 10(-6) dyn s/cm. These values are in relatively good agreement with the values calculated from lateral and rotational mobility measurements.