  We prove that entire solutions of the minimal hypersurface equation \[
  \mathrm{div}\left(\frac{Du}{\sqrt{1+|Du|^2}}\right) = 0 \] on a complete
manifold with $\mathrm{Ric} \ge 0$, whose negative part grows like
$\mathcal{O}(r/\log r)$ ($r$ the distance from a fixed origin), are constant.
This extends the Bernstein Theorem for entire positive minimal graphs
established in recent years. The proof depends on a new technique to get
gradient bounds by means of integral estimates, which does not require any
further geometric assumption on $M$.
