Let R be a Noetherian integral domain which is graded by an ordered group G and let x be a set of n variables with a term order. It is shown that a finite subset F of R[x] is a weak (respectively strong) Grobner basis in R[x] graded by G x Z^n if and only if F is a weak Grobner basis in R[x] graded by {0} x Z^n and certain subsets of the set of leading coefficients of the elements of F form weak (respectively strong) Grobner bases in R. It is further shown that any G-graded ring R$ for which every ideal has a strong Grobner basis is isomorphic to k[x_1,...,x_n], where k is a PID.