  We observe and study the phenomenon of Anderson localization in a system of
true quantum kicked rotors. Nitrogen molecules in a supersonic molecular jet
are cooled down to 27~K and are rotationally excited by a periodic train of
24~high-intensity femtosecond pulses. Exponential distribution of the molecular
angular momentum - the most unambiguous signature of Anderson localization - is
measured directly by means of coherent Raman scattering. We demonstrate the
suppressed growth of the molecular rotational energy with the number of laser
kicks and study the dependence of the localization length on the kick strength.
Both timing and amplitude noise in the pulse train is shown to destroy the
localization and revive the diffusive growth of angular momentum.
