We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic partial equations of the following form: $D_t^αu(t, x)=\textit{B}u+u\cdot W^H$, where $D_t^α$ is the fractional derivative of order $α$ with respect to the time variable $t$, $\textit{B}$ is a second order elliptic operator with respect to the space variable $x\in\mathbb{R}^d$, and $W^H$ a fractional Gaussian noise of Hurst parameter $H=(H_1, \cdots, H_d)$. We obtain conditions satisfied by $α$ and $H$ so that the square integrable solution $u$ exists uniquely .