[en] The elastic proton-helium differential cross section has been determined for incident laboratory energies from 45 to 400 GeV in the range 0.003< or =Vertical BartVertical Bar< or =0.52 (GeV/c)² by means of the internal-gas-jet-target technique. The differential cross section drops 4 --5 orders of magnitude to the first dip at Vertical BartVertical Barapprox. =0.22 (GeV/c)². The shrinkage in the slope of the differential cross section is found to be twice as fast as that in the proton-proton case. The slope parameter at Vertical BartVertical Barapprox. =0 is described by the formula b = 24+1.13 ln s, where b is in (GeV/c)^-2 and s is in GeV². The elastic proton-helium cross section is normalized to the known elastic proton-proton cross section using data taken with a helium and hydrogen mixture as a target. The proton-helium total cross section is determined from the optical theorem. The total cross section rises by 4% between 100 and 400 GeV. Results are presented on the real part of the elastic-scattering amplitude and on the total elastic cross section. The experimental differential cross sections are compared to Glauber-model predictions