The nonlinear localized vibrational modes in an anharmonic atomic chain with uniform mass but two alternating force constants between nearest-neighbors are stud ied by means of multiple-scale expansion. This atomic chain models the vibration s of an arrow of atoms in the direction of a diamond-structure lattice or a molecular chain. It is shown that the distribution of the atomic displacements is governed by a perturbed nonlinear Schrdinger equation, and both the statio nary and moving solutions are obtained. The results are somewhat different from that of the diatomic chain with uniform force constants but two alternating mass es. The reason may be that the translational symmetry of diamond-structure latti ce is comparetively lower.