Inelastic neutron-scattering experiments on the high-temperature superconductor La1.855Sr0.145CuO4 reveal a magnetic excitation gap Delta that decreases continuously upon application of a magnetic field perpendicular to the CuO2 planes. The gap vanishes at the critical field required to induce long-range incommensurate antiferromagnetic order, providing compelling evidence for a field-induced soft-mode driven quantum phase transition.