Magnetic properties of the one-dimensional spin - orbital model are investigated in a dimer - singlet phase. By means of the density matrix renormalization group method, we calculate the magnetization curve and the correlation functions in a magnetic field. It is found that a cusp singularity appears in the magnetization process when the ratio of the exchange couplings is properly tuned. The origin of the cusp singularity is discussed. (C) 2003 Elsevier B. V. All rights reserved.