We study the abelian period sets of Sturmian words, which are codings of irrational rotations on a one-dimensional torus. The main result states that the minimum abelian period of a factor of a Sturmian word of angle α with continued fraction expansion [0; a₁, a₂, ...] is either tq_k with 1 ≤ t ≤ a_k+1 (a multiple of a denominator q_k of a convergent of α) or q_k,l (a denominator q_k,l of a semiconvergent of α). This result generalizes a result of Fici et al. stating that the abelian period set of the Fibonacci word is the set of Fibonacci numbers. A characterization of the Fibonacci word in terms of its abelian period set is obtained as a corollary.