Recent investigations of the superconducting iron-arsenide families have highlighted the role of pressure, be it chemical or mechanical, in fostering superconductivity. Here we report that CaFe2As2 undergoes a pressure-induced transition to a non-magnetic, volume collapsed tetragonal phase, which becomes superconducting at lower temperature. Spin-polarized total-energy calculations on the collapsed structure reveal that the magnetic Fe moment itself collapses, consistent with the absence of magnetic order in neutron diffraction.