An exchange functional is proposed which is conceptually similar to the established Becke functional and satisfies the same asymptotic properties. The one-parameter variant A of this new functional yields atomic exchange energies which are very close to those from the optimized potential model (OPM) for the light elements (up to Ne), whereas the three-parameter variant B reproduces the OPM reference values for all elements (H to Xe) more accurately than other established exchange functionals. For molecular applications, the new exchange functional is combined with a new non-local correlation functional based on a Coulomb hole model. Detailed results are reported for the molecules from the G2 data base. For each of the four basis sets considered, variants A and B yield essentially the same atomization energies, ionization potentials, proton affinities and molecular geometries. The mean absolute deviation between the calculated and experimental atomization energies for the G2 reference molecules is 3 kcal mol-1. In an overall assessment, the present results seem comparable in quality with those from the B3LYP hybrid functional, even though our new functional does not include any Hartree-Fock exchange. Some illustrative results for transition-metal compounds are also given.