Let (A,m) be a local Noetherian ring, and let P be prime ideal of A whose height and analytic spread are equal. Let R(P) be the Rees ring \(\oplus P^ n,\quad n\geq 0\). The authors consider what the Gorensteinness of R(P) implies about R and P. Theorem: If (A,m) is a generalized Cohen- Macaulay ring (i.e., a ring of finite local cohomology) with Dim(A)\(\geq 4\), if height(P) \(= analytic\) spread \(of\quad P =2,\) and if R(P) is Gorenstein, then P is generated by a regular sequence, and A is Gorenstein. In the case that \(Dim(A)=3\), the authors conjecture that the theorem remains true, and prove it in cases in which the multiplicity of A is small. The final section of the paper treats the case that \(P=m\).