STARS form inside dense clouds of molecular gas, but the details of the process, such as the quantity of gas that goes into stars and the rate at which the gas collapses, are still unknown. The earliest stages of cloud collapse are particularly poorly understood; some theoretical models exist(1-4), but there has been no observational evidence to support them. Here we report molecular emission-line data from the Taurus molecular cloud, which allows us to follow the earliest stages of cloud collapse. We find, contrary to previous results(5), that the cloud cores without young stars are less dense and more extended than those with stars, and that the timescale of core collapse suggested by the data is a few hundred thousand years. This is in good agreement with a model in which the formation rate of low-mass stars is controlled by ambipolar diffusion-the relative drift of neutral molecules with respect to magnetic field lines in the cloud(1,3).