As it is well known, an embedded minimal hypersurface in \(R^{n+1}\) with an isolated singularity may be obtained by taking any minimal hypersurface \(\Sigma\) in the n-sphere \(S^ n\), other than a great hemisphere, and forming the one cover \(\Sigma\), \(C=\{rx| x\in \Sigma\), \(0<r<\infty \}\). The question arises whether there exists an embedded minimal hypersurface in \(R^{n+1}\) which has one isolated singularity, but which is not a cone. In this paper, such examples are found for \(n\geq 3\). Moreover, each example constructed here is asymptotic to a given, completely arbitrary, nonplanar minimal cone and is stable in case the cone satisfies a strict stability inequality. Note that this is trivially false for \(n=1\). For \(n=2\), the question remains open.