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We derive the normal detonation shock velocity-curvature relation for a near-Chapman-Jouguet detonation, for an explosive material with Arrhenius kinetics and a large activation energy. Large activation energy asymptotics are used to develop an explicit exponential formula relating the shock curvature κ to the normal detonation shock speed, D_n. In this case, the D_n-κ relation is multivalued and has a turning point with a critical curvature κ_cr such that for κ > κ_cr, the possibility of detonation extinction arises. The asymptotic formula is in excellent agreement with the exact solution found by a numerical shooting procedure.