The author proves that there exists, up to a homothety, a unique invariant Einstein metric on any five-dimensional homogeneous space of the form \(SU(2)\times SU(2)/SO(2)_ r\), where \(r\) is a rational number such that \(| r|\neq 1\), and \(SO(2)_ r\) denotes the subgroup of all product matrices of the form \[ \begin{pmatrix} e^{2\pi it}& 0\\0 &e^{- 2\pi it}\end{pmatrix}\times\begin{pmatrix} e^{2\pi irt} &0\\0 &e^{-2\pi irt}\end{pmatrix}\qquad (t\in\mathbb{R}). \] These metrics are never naturally reductive.