[en] For some time the York time parameter has been identified as a candidate for a physically meaningful time in cosmology. An associated Hamiltonian may be found by solving the Hamiltonian constraint for the momentum conjugate to the York time variable, although an explicit solution can only be found in highly symmetric cases. The Poisson structure of the remaining variables is not canonical. Here we quantise this dynamics in an anisotropic minisuperspace model via a natural extension of canonical quantisation. The resulting quantum theory has no momentum representation. Instead the position basis takes a fundamental role. We illustrate how the quantum theory and the modified representation of its momentum operators lead to a consistent theory in the presence of the constraints that arose during the Hamiltonian reduction. The quantised reduced Hamiltonian is Hermitian, although the momentum operators are not, the causes and implications of which we discuss. We are able to solve for the eigenspectrum of the Hamiltonian. Finally we discuss how far the results of this model extend to the general non-homogeneous case, in particular perturbation theory with York time. (paper)