Effective collision strengths for the 10 astrophysically important fine-structure forbidden transitions among the S-4 degrees, D-2 degrees and P-2 degrees levels in the 3s(2)3p(3) configuration of Cl III are presented. The calculation employs the multichannel R-matrix method to compute the electron-impact excitation collision strengths in a close-coupling expansion, which incorporates the lowest 23 LS target eigenstates of Cl III. These states are formed from the 3s(2)3p(3), 3s3p(4), 3s(2)3p(2)3d and 3s(2)3p(2)4s configurations. The Maxwellian-averaged effective collision strengths an presented graphically for all 10 fine-structure transitions over a wide range of electron temperatures appropriate for astrophysical applications [log T(K) = 3.3 - log T(K) = 5.9]. Comparisons are made with the earlier seven-state close-coupling calculation of Butler & Zeippen, and in general excellent agreement is found in the low-temperature region where a comparison is possible [log T(K) = 3.3 - log T(K) = 4.7], However, discrepancies of up to 30 per cent are found to occur for the forbidden transitions which involve the S-4 degrees ground state level, particularly for the lowest temperatures considered. At the higher temperatures, the present data are the only reliable results currently available.