  For a class of orthogonal polynomials related to the $q$-Meixner polynomials
corresponding to an indeterminate moment problem we give a one-parameter family
of orthogonality measures. For these measures we complement the orthogonal
polynomials to an orthogonal basis for the corresponding weighted $L^2$-space
explicitly. The result is proved in two ways; by a spectral decomposition of a
suitable operator and by direct series manipulation. We discuss extensions to
explicit non-positive measures and the relation to other indeterminate moment
problems for the continuous $q^{-1}$-Hahn and $q$-Laguerre polynomials.
