The radio frequency identification (RFID) technology has been widely adopted and being deployed as a dominant identification technology in a health care domain such as medical information authentication, patient tracking, blood transfusion medicine, etc. With more and more stringent security and privacy requirements to RFID based authentication schemes, elliptic curve cryptography (ECC) based RFID authentication schemes have been proposed to meet the requirements. However, many recently published ECC based RFID authentication schemes have serious security weaknesses. In this paper, we propose a new ECC based RFID authentication integrated with an ID verifier transfer protocol that overcomes the weaknesses of the existing schemes. A comprehensive security analysis has been conducted to show strong security properties that are provided from the proposed authentication scheme. Moreover, the performance of the proposed authentication scheme is analyzed in terms of computational cost, communicational cost, and storage requirement.