Photoionization cross sections for the 3p and 3s subshells of 3s'3p' 'P ground state of atomic chlorine are calculated in the photon energy range from the Cl+ 3p 'P ionization threshold to 65 eV using the R-matrix method. The initial Cl (3s 3p' P') bound state, the final Cl+ plus photoelectron ( S, P, D) states, and the residual Cl+ (3p P, 'D, 'S, 3s 3p ' P, 'P ) states are represented by extensive configuration-interaction wave functions. The photoionization cross sections are dominated by Rydberg series of resonances in the threshold region. The present results are compared with the available experimental measurements and other calculations. PACS number(s): 32.80.Fb, 32.80.0z