This paper reports the first experimental observation and theoretical description of zero time lag, complete-chaos synchronization using three laser diodes coupled in a unidirectional cascade configuration. The master laser is rendered chaotic by optical feedback from an external mirror. The output intensity of the master laser is injected into an intermediate laser, and the output intensity of this intermediate laser is in turn injected into a slave laser. The distances between the lasers are carefully adjusted such that the times of flight between the lasers are equal to the delay time in the external-cavity of the master laser. When the three lasers are subject to the same optical injection strength it is observed that their output intensities synchronize with zero time lag. The chaos synchronization has been confirmed by time traces and synchronization diagrams of the three laser outputs, and the null lag time has been measured from cross-correlation diagrams. Numerical simulations based on single-mode rate equations are in good qualitative agreement with the experimental observations.