A new algorithm for the calculation of molecular integrals involving STOs is reported. The algorithm enables us to obtain every two-center one-electron integral and the long-range many-center one- and two-electron integrals. The efficient implementation of the algorithm is discussed and its performance is thoroughly tested. The analysis on the stability of the relations employed in the calculation of multipolar moments is included. Futhermore, the computer time required to carry out each step (construction of basic matrices, calculation of multipolar moments, and calculation of two-electron integrals) has also been analyzed. The range of validity of this approach is shown in several molecular integrals.