Numerical calculations of the dynamics of a spherically symmetric collapsing proto-star of one solar mass have been made for various initial conditions. Calculations have also been made for masses of 2M0 and M o. In all cases the collapse is found to be extremely non-homologous and is such that a very small part of the cloud's mass at the centre reaches stellar densities and stops collapsing before most of the cloud has had time to collapse very far. The stellar core thus formed subsequently grows in mass as material falls into it, finally becoming an ordinary star when all of the proto-stellar material has been accreted. During most of this time the stellar core is completely obscured by the dust in the infalling cloud, the absorbed radiation reappearing in the infra-red as thermal emission from the dust grains. The resulting star is almost a conventional Hayashi pre-main sequence model, but it appears rather low on the Hayashi track. For masses much greater than about 2M0 the convective Hayashi phase does not exist at all. It appears that certain properties of T Tauri stars may find explanation in the results of the present calculations. In an appendix to the paper it is shown that limiting forms may be derived for the density and velocity distributions near the centre of an isothermally collapsing sphere. This may be shown to be possible also for a sphere with a polytropic equation of state. Numerical results are presented for the limiting solution in the isothermal case.