Cross sections for the coherent photoproduction of $\rho^{0}$ vector mesons in ultra-peripheral Pb-Pb collisions at $ \sqrt{s_{\mathrm{NN}}}= 5.02$ TeV are reported. The measurements, which rely on the $\pi^+\pi^-$ decay channel, are presented in three regions of rapidity covering the range $|y|<~0.8$. For each rapidity interval, cross sections are shown for different nuclear-breakup classes defined according to the presence of neutrons measured in the zero-degree calorimeters. The results are compared with predictions based on different models of nuclear shadowing. Finally, the observation of a coherently produced resonance-like structure with a mass around 1.7 GeV/$c^2$ and a width of about 140 MeV/$c^2$ is reported and compared with similar observations from other experiments.