The Schr\"odinger equation is considered on the half line with a selfadjoint
boundary condition when the potential is real valued, integrable, and has a
finite first moment. It is proved that the potential and the two boundary
conditions are uniquely determined by a set of spectral data containing the
discrete eigenvalues for a boundary condition at the origin, the continuous
part of the spectral measure for that boundary condition, and a subset of the
discrete eigenvalues for a different boundary condition. This result provides a
generalization of the celebrated uniqueness theorem of Borg and Marchenko using
two sets of discrete spectra to the case where there is also a continuous
spectrum. The proof employed yields a method to recover the potential and the
two boundary conditions, and it also constructs data sets used in various
inversion methods. A comparison is made with the uniqueness result of Gesztesy
and Simon using Krein's spectral shift function as the inversion data.