We demonstrate quantum phase engineering with two novel techniques that allow us to write and read out the spatial phase distribution of a Bose-Einstein condensate (BEC). A quantum state is designed and produced by optically imprinting a phase pattern onto a BEC of sodium atoms, and matter-wave interferometry with spatially resolved imaging is used to analyze the resultant phase distribution. An appropriate phase imprint creates solitons, the first experimental realization of this nonlinear phenomenon in a BEC. We investigate the subsequent evolution of these excitations both experimentally and theoretically. We also use the interferometer to measure the quantum phase evolution of a BEC that has been released from a magnetic trap.