Differential cross sections for He3(p,p)He3 elastic scattering have been measured at 11 energies in the laboratory energy range 19.5 to 47.5 MeV. The most forward c.m. angle for the angular distributions varies from 10.1°to 13.4°, and the most backward angle varies from 163.2°to 173.4°. The relative errors in the data are usually less than 2%, and the scale error is 1.5%. The present data, together with analyzing power and total reaction cross section data of others, have been subjected to an energy-dependent phase shift analysis. The extracted phase shifts and the differential cross sections are compared with the results of a simple resonating-group calculation.