Let p be a prime and let J1(pn) denote the Jacobian of the modular curve X1(pn). The Jacobian J1(pn) contains a ℚ-rational torsion subgroup generated by the cuspidal divisor classes [(a/pn)−(∞)], where p∤ a. In this paper, we determine the structure of the p-primary subgroup of this ℚ-rational torsion subgroup in the case where p is a regular prime.