A locally convex space E is called distinguished if its strong dual \((E',\beta (E',E))\) is barrelled. In 1954, A. Grothendieck posed the problem, whether the bidual of a distinguished Fréchet space is again distinguished. In this note we give a negative answer to that question. We make use of the fact that Fréchet spaces of Moscatelli type of the form \[ E=\{(y_ k)_{k\in {\mathbb{N}}}\in Y^{{\mathbb{N}}}:(f(y_ k))_{k\in {\mathbb{N}}}\in c_ 0(X)\}, \] where Y,X are Banach spaces and f: \(Y\to X\) a continuous linear map, are always distinguished, and we prove that E has a distinguished bidual if and only if f is open onto its range. More generally, we prove that Fréchet spaces of the form \[ F=\{(y_ k)_{k\in {\mathbb{N}}}\in Y^{{\mathbb{N}}}:(f(y_ k))_{k\in {\mathbb{N}}}\in \ell^{\infty}(X)\} \] are distinguished if and only if they are quasinormable.