Random heteropolymers (RHPs) with uncorrelated sequence fluctuations on the segmental scale can undergo a transition wherein, below a certain temperature, the thermodynamics is determined by a few dominant conformations. We study this "freezing" transition for RHPs with correlated sequence fluctuations. Specifically, we apply our theory to the case where the correlations decay with a single correlation length; a pragmatically realizable example is provided by random block copolymers. Our results show that the temperature at which freezing occurs grows with the block length of such polymers. Freezing also occurs on the scale of the correlation length, thus making experimental observation of this phenomenon (a consequence of frustration coupled with quenched disorder) more accessible. The results are rationalized on physical grounds.