The Clausing integral equation for free-molecular gas flow in a cylindrical tube of arbitrary length is solved by an efficient numerical method. The current numerical results improve upon the precision of values for the molecular transmission probabilities (molecular flow rates or Clausing factors) reported previously in the literature. The basic approach involves the use of a singularity subtraction technique to smooth the kernel of the Clausing integral equation, which is a Fredholm integral equation of the second kind, before solving it via the use of Gauss-Legendre quadratures. The singularity subtraction technique is further modified by the additional subtraction of Clausing's long tube solution prior to the use of the Gauss-Legendre quadratures. This modification has been shown to improve the speed of the computations, the rate at which the results converge, and the precision of the results, particularly for medium length tubes and in the long tube limit. (c) 2007 American Vacuum Society.