In this article, we consider the group F ∞ 1 (N) of modular units on X 1 (N) that have divisors supported on the cusps lying over ∞ of X 0 (N), called the ∞-cusps. For each positive integer N, we will give an explicit basis for the group F ∞ 1 (N). This enables us to compute the group structure of the rational torsion subgroup C ∞ 1 (N) of the Jacobian J 1 (N) of X 1 (N) generated by the differences of the ∞-cusps. In addition, based on our numerical computation, we make a conjecture on the structure of the p-primary part of C ∞ 1 (p n ) for a regular prime p.