Yes. And so do the faithful in Eastern Churches. Only the Latin Church offers its faithful the host but no chalice. Don't ask me why. As a Catholic.
That practice, as far as I know, only became popular in certain American dioceses that also, completely coincidentally of course, engaged in post Vatican II nonsense including guitar Masses, pro abortion nuns, liberation theology preachers. It is either entirely corrected or nearly corrected as BXVI imposes orthodoxy upon the Augean stables that house the USCCB.