Monocular depth foundation models make it tempting to replace depth sensors on mobile robots with a single RGB camera. We show that, used off the shelf, this is unsafe: in a simulated indoor navigation benchmark, a standard Nav2 stack driven by Depth Anything V2 (metric) collides in 63\% of 120 episodes, because the model systematically over-estimates the distance to nearby obstacles. We propose a lightweight plug-in module that turns predicted depth into \emph{safe depth}. A 0.58\,M-parameter head on top of the \emph{frozen} depth model predicts, per pixel, a log-depth correction $μ$ and an uncertainty $σ$, trained with a $β$-weighted Gaussian negative log-likelihood. The navigation stack consumes the one-sided bound $\hat d_{\text{safe}}=d\,e^{μ-kσ}$, where $k$ is set by conformal calibration on a held-out world, and uncertainty also decides which pixels may be treated as floor. The correction halves the depth error (AbsRel $0.262\rightarrow0.134$) and reduces near-range over-estimation from 59.8\% to 12.0\%. In navigation, safe depth raises the success rate from 37\% to 70\% and lowers collisions from 63\% to 3\%, close to the 2\% of ground-truth depth. A constant safety margin calibrated to the same pixel coverage reaches only 30\% success because it freezes the robot; uncertainty-aware safety wins 52 of 56 discordant paired episodes against it. Gains hold on glass, texture-less walls and dark regions.