We give a new simplification method, called E-cycle Simplification, for Basic Completion inference systems. We prove the completeness of Basic Completion with E-cycle Simplification. We prove that E-cycle Simplification is strictly stronger than the only previously known complete simplification method for Basic Completion, Basic Simplification, in the sense that every derivation involving Basic Simplification is a derivation involving E-cycle Simplification, but not vice versa. E-cycle Simplification is simple to perform, andq does not use the reducibility-relative-to condition. ECC implements our method.