The exact solution of the asymmetric exclusion problem with first- and scond-class particles is presented. In this model the particles (size 1) of both classes are located at lattice points, and diffuse with equal asymmetric rates, but particles in the first class do not distinguish those in the second class from holes (empty sites). We generalize and solve exactly this model by considering molecules in the first and second class with sizes s1 and s2 (s1, s2 = 0, 1, 2, ...), in units of lattice spacing, respectively. The solution is derived by a Bethe ansatz of nested type. We give a simple pedagogical presentation of the Bethe ansatz solution of the problem which can easily be followed by a reader not specialized in exactly integrable models.