The axial dispersion coefficient in a fluid in laminar flow in a tube is generally smaller in a curved tube than in a straight tube, because of the enhacement of lateral transport by secondary flows. In this paper the extent of this reduction is computed using Horn's modification of Aris's method of moments. Dimensional analysis suggests that the most convenient form in which to represent the results is that obtained empirically by Trivedi and Vasudeva. The results presented here cover the whole laminar flow regime, and show the region of applicability of previously computed dispersion coefficients to be limited. The computed dispersion coefficients agree well with previously reported experimental results for small Schmidt numbers; however, discrepancies are present at large Schmidt numbers. Possible causes are discussed.