The Cl35(n,p) and Cl35(n,α) cross sections at incident neutron energies between 2.42 and 2.74 MeV were measured using the Berkeley High Flux Neutron Generator. The cross sections for Cl35(n,p) were more than a factor of 3 to 5 less than all of the values in the neutron absorption data libraries, while the Cl35(n,α) cross sections are in reasonable agreement with the data libraries. The measured energy-differential cross section is consistent with a single resonance with a width of 293(46) keV. This result suggests that, despite the high incident neutron energy, any attempt to model (n,x) cross sections in the vicinity of the N=Z=20 shell gap requires a resolved resonance approach rather than a Hauser-Feshbach approach.