A promising "underlying bivariate normal" approach was proposed by Jöreskog and Moustaki for use in the factor analysis of ordinal data. This was a limited information approach that involved the maximization of a composite likelihood function. Its advantage over full-information maximum likelihood was that very much less computation was involved. Little statistical and computational information was provided for its application in practice. The aim of this article is to provide statistical and computational methodology to enable the Jöreskog and Moustaki's approach to be routinely applied in the factor analysis of ordinal data. A constrained pseudo Fisher-scoring algorithm for parameter estimation is developed and is implemented in a computer program written in FORTRAN 95. This algorithm imposes inequality constraints to ensure that all estimates are admissible. Statistical properties of the approach are considered and illustrated by means of simulation studies and real data examples.