Cooperative Adaptive Cruise Control (CACC) has the potential to significantly improve traffic throughput by reducing inter-vehicle following gap. However, communication and actuation delays in CACC systems are detrimental to string stability. A much larger time gap than the lump sum of the two delays is often required for CACC to guarantee string stability, which compromises their benefits on traffic throughput. It remains as a challenge to design string-stable CACC controllers that operate at small time gaps under delays. In this paper, we design a memory-anticipation CACC to explicitly handle communication and actuation delays to fulfill string stability specifications. The proposed controller employs a novel memory-anticipation spacing policy to address communication and actuation delays and is applicable to heterogeneous platoons. Unlike existing CACC controllers, our proposal utilizes past information of the preceding vehicle (i.e. the memory) instead of its current state/control information, which makes the controller robust to communication delay. Furthermore, a Smith predictor (i.e. the anticipation) is designed to compensate for the actuation delay. Both individual vehicle stability and string stability can be guaranteed with the minimum time gap as small as the summation of communication delay and actuation delay. Controller analysis and simulation verify the enhancement of the stability performance with respect to existing studies.