We demonstrate that intense laser pulses can be used to directly control the spins in ferrimagnetic garnet films. Through an ultrafast and nonthermal photomagnetic effect the magnetocrystalline anisotropy is modified to create a new long-lived equilibrium orientation for the magnetization. Simultaneously, the magnetization is rotated into this new state by precession in a strong transient optically generated magnetic field. All take place within the 100 fs duration of a single laser pulse, thus demonstrating the feasibility of photomagnetic switching on the femtosecond time scale.