A method of accelerating convergence is discussed on a numerical computation of the normal distribution function. It is shown that modifying the Laplace continued fraction for the normal integral Q(u) by using appropriate accelerative tails yields convergence acceleration with a remarkable reduction of the number of terms required for a given accuracy. In addition sequences of upper and lower bounds for Q(u) are also derived and hence in a wide region of parameters new modified continued fractions discussed here are proved to be useful at a higher level of accuracy as well as at a lower level.