A physical scheme for teleporting unknown atomic entangled states via three-atom non-maximally entangled states is proposed in cavity QED. The distinct advantages of the scheme are that the teleportation and distillation procedure can be realized simultaneously, and the scheme is insensitive to the cavity decay and thermal field with the assistance of a strong classical driving field. It is also unnecessary to concentrate non-maximally entanglement state quantum channel in advance. In addition, the joint Bell-state measurement can be distinguished via detecting the atomic state.