Chiral edge states are the hallmark of two-and three-dimensional topological materials, but their one-dimensional (1D) analog has not yet been found. We report that the 1D topological edge states, solitons, of the charge density wave system of indium atomic wires self-assembled on a silicon surface have chirality. The system is described by a coupled double Peierls-dimerized atomic chain, where the interchain coupling induces dynamical sublattice symmetry breaking. This changes its topological symmetry from Z(2) x Z(2) to Z(4) and endows solitons with a chiral degree of freedom. Chiral solitons can produce quantized charge transport across the chain that is topologically protected and controllable by the soliton's chirality. Individual right-and left-chiral solitons in indium wires are directly identified by scanning tunneling microscopy.