  We solve the path integral in momentum space for a particle in the field of
the Coulomb potential in one dimension in the framework of quantum mechanics
with the minimal length given by
  $(\Delta X)_{0}=\hbar \sqrt{\beta}$, where $\beta$ is a small positive
parameter. From the spectral decomposition of the fixed energy transition
amplitude we obtain the exact energy eigenvalues and momentum space
eigenfunctions.
