The second author defined in [\textit{C. Olmos}, J. Differ. Geom. 39, 605-627 (1994; Zbl 0806.53054)] the rank of a homogeneous submanifold of a Euclidean space as the maximal number of locally defined and linearly independent parallel normal vector fields. He proved in the same paper that if the rank of an irreducible homogeneous submanifold is at least two, then it is the orbit of the isotropy representation of a symmetric space. In the paper under review, the isoparametric rank of an arbitrary submanifold is defined. The main result is that a locally irreducible submanifold with isoparametric rank at least two has constant principal curvatures.