Motivated by recent nuclear magnetic resonance experiments on ZnCu3(OH)(6)Cl-2, we present an exact-diagonalization study of the combined effects of nonmagnetic impurities and Dzyaloshinskii-Moriya (DM) interactions in the s = 1/2 kagome antiferromagnet. The local response to an applied field and correlation-matrix data reveal that the dimer freezing which occurs around each impurity for D = 0 persists at least up to D/J similar or equal to 0.06, where J and D denote, respectively, the exchange and DM interaction energies. The phase transition to the (Q = 0) semiclassical 120 degrees state favored at large D takes place at D/J similar or equal to 0.1. However, the dimers next to the impurity sites remain strong up to values D similar to J, far above this critical point, and thus do not participate fully in the ordered state. We discuss the implications of our results for experiments on ZnCu3(OH)(6)Cl-2.