We show that the classical stochastic motion of an open bosonic string leads to the same results as the standard first quantization of this system. For this, the diffusion constant governing the process has to be proportional to alpha', the Regge slope parameter, which is the only constant, along with the velocity of light, needed to describe the motion. (C) 1999 Published by Elsevier Science B.V. All rights reserved.256417008994