The ALICE Collaboration reports the first measurement of the production of prompt $Ξ_{\mathrm c}^{0}$ baryons in nucleus$-$nucleus collisions by analyzing data from $\mathrm{Pb}-\mathrm{Pb}$ collisions at $\sqrt{s_{\mathrm{NN}}}~=5.02~\mathrm{TeV}$. The production and transverse-momentum ($p_{\mathrm T}$) differential spectra of $Ξ_{\mathrm c}^{0}$ are particularly sensitive to the hadronization process and to strangeness production in the quark--gluon plasma formed in high-energy heavy-ion collisions. The $Ξ_{\mathrm c}^{0}$ baryons are reconstructed at midrapidity ($|y|<~0.5$) in the transverse-momentum intervals $3<~p_{\mathrm T}~<~12~\mathrm{GeV}/c$ and $4<~p_{\mathrm T}~<~12~\mathrm{GeV}/c$ in the 0$-$10\% and 30$-$50\% centrality intervals, respectively. The nuclear modification factor ($R_{\mathrm{AA}}$) reaches values up to 3 in the interval $3<~p_{\mathrm T}~<~4~\mathrm{GeV}/c$, which is the largest value measured so far for charm hadrons. Model predictions are compatible with the measured $R_{\mathrm{AA}}$, while, in the measured $p_{\mathrm T}$ intervals, they underestimate the production yield as well as the measured $Ξ_{\mathrm c}^{0}/\mathrm{D}^{0}$, $Ξ_{\mathrm c}^{0}/Λ_{\mathrm c}^{+}$, and $Ξ_{\mathrm c}^{0}/\mathrm{D}^{+}_{\mathrm s}$ yield ratios.