Let \(A\) be a finite-dimensional simple (associative) algebra over the field \(\mathbb{Q}\) of rational numbers. By an order of \(A\), we mean a subring \(R\) of \(A\), which contains the unit of \(A\) and forms a lattice in \(A\) (viewed as a vector space over \(\mathbb{Q}\)). It is well-known that orders exist, each one is contained in a maximal order of \(A\), and maximal orders share a common discriminant (called discriminant of \(A\)). The paper under review provides an algorithm for constructing a maximal order of \(A\). As a first step, the algorithm determines the decomposition of \(A\) into a direct product of simple algebras \(A_1,\dots,A_s\) (and solves the corresponding problem concerning the centre of \(A\)). The second step is to compute consecutively the discriminants and the local Schur indices of \(A_i\), \(i=1,\dots,s\). On this basis, a maximal order \(M_i\) of \(A_i\) is constructed, for each index \(i\), as a result of a repeated application of a canonical process used by \textit{H. Benz} and \textit{H. Zassenhaus} [in J. Number Theory 20, 282-298 (1985; Zbl 0593.16005)]; the subring \(M\) of \(A\) generated by \(M_i\), \(i=1,\dots,s\), is a maximal order of \(A\). Compared with the general case, the computations needed to implement the algorithm are easier in the special case where \(A\) has uniformly distributed invariants, e.g., when \(A\) is the centralizer algebra of a finite group representation.