Investigations of the onset of superconductivity in ultra-thin films of Ga and Pb, deposited onto substrates held at low temperatures have been carried out. The measurements have revealed the existence of a threshold for superconductivity in which the normal-state sheet resistance is the only relevant parameter. It is found that only when the normal-state sheet resistance falls below h/4e2 does a film exhibit zero resistance at low temperatures. This value for the threshold is in agreement with recent theoretical models predicting a dissipation driven phase transition. At values greater than this threshold, but nevertheless close to it, a systematic and rapid variation of the low-temperature resistance with normal-state resistance is observed. We interpret this behavior as due to strong zero-point quantum fluctuations and speculate that it can be modelled as a 2-D XY transition in which dissipation rather than temperature is the control -parameter.