  Let $\frak{n}$ be a square-free ideal of $\mathbb{F}_q[T]$. We study the
rational torsion subgroup of the Jacobian variety $J_0(\frak{n})$ of the
Drinfeld modular curve $X_0(\frak{n})$. We prove that for any prime number
$\ell$ not dividing $q(q-1)$, the $\ell$-primary part of this group coincides
with that of the cuspidal divisor class group. We further determine the
structure of the $\ell$-primary part of the cuspidal divisor class group for
any prime $\ell$ not dividing $q-1$.
