We study the localization length l(c) of a pair of two attractively bound particles moving in a one-dimensional random potential. We show in which way it depends on the interaction potential between the constituents of this composite particle. For a pair with many bound states N the localization length is proportional to N, independently of the form of the two particle interaction. For the case of two bound states, we present an exact solution for the corresponding Fokker-Planck equation and demonstrate that l(c) depends sensitively on the shape of the interaction potential and the symmetry of the bound state wave functions. (C) 2003 Elsevier Science B.V. All rights reserved.