  We prove the existence and uniqueness up to translations of the solution to a
Minkowski type problem for the torsional rigidity in the class of open bounded
convex subsets of the $n$-dimensional Euclidean space. For the existence part
we apply the variational method introduced by Jerison for analogous problems
concerning other variational functionals. Uniqueness follows from the
Brunn--Minkowski inequality for the torsional rigidity and corresponding
equality conditions.
