  In this paper we consider modification of general relativity extending $R - 2
\Lambda$ by nonlocal term of the form $\sqrt{R-2\Lambda}\, \mathcal{F}(\Box)\,
\sqrt{R-2\Lambda} ,$ where $\mathcal{F}(\Box)$ is an analytic function of the
d'Alembert operator $\Box$. We have found some exact cosmological solutions of
the corresponding equations of motion without matter and with $\Lambda \neq 0$.
One of these solutions is $ a (t) = A \, t^{\frac{2}{3}} \,
e^{\frac{\Lambda}{14} t^2} ,$ which imitates properties similar to an interplay
of the dark matter and the dark energy. For this solution we calculated some
cosmological parameters which are in a good agreement with observations. This
nonlocal gravity model has not the Minkowski space solution. We also found
several conditions which function $\mathcal{F}(\Box)$ has to satisfy.
