We extend the two-center wave-packet convergent close-coupling method to proton collisions with helium. The target is treated as a three-body system, where correlations between the electrons are taken into account. We apply a frozen-core approximation, where one of the electrons is described by the He+1s orbital, and obtain the helium singlet wave functions as well as the energy levels using a numerical approach. The wave-packet approach is used to discretize the continuum of the target and the hydrogen atom formed after electron capture by the projectile. Convergence of the results is studied in terms of the included projectile- and target-centered states. We present electron-capture and single- and double-ionization cross sections for protons incident on He in the ground state in the energy range from 15 keV to 1 MeV. We also provide partial cross sections for electron capture and direct excitation into the n=2 shell states of hydrogen and helium, respectively. Our results are in good agreement with available experimental data and other calculations, where available.