Spectral properties of a self-adjoint operator corresponding to a limit of $-\Delta +\xi _{\varepsilon}+c_{\varepsilon}$ as $\varepsilon \to 0$ are invetigated, where $\Delta$ is the Laplacian  on ${\mathbb R}^2$,
$\xi _{\varepsilon}$ is a smooth approximation of the white noise $\xi$ defined by $\exp (\varepsilon ^2\Delta )\xi$, and $c_{\varepsilon}$ is a positive function of  $\varepsilon$ diveging as $\varepsilon \to 0$.
For sufficiently low energies, it is proven that phenomena of the Anderson localization occur: the spectrum is pure point and the corresponding eigenfunctions decay exponentially at infinity.
For the proof, the Wegner estimate and the multi scale analysis are modified appropriately.