Let \(X\) be a smooth projective Calabi-Yau threefold, \(Y\) be a normal Calabi-Yau variety with only ordinary double points as singularities and \( f:X\rightarrow Y\) be a small contraction morphism. If \(\mathcal{L}_{0}\) is an ample line bundle over \(Y\) and \(\alpha \) is a Kähler class on \(X\), then by a familiar theorem by Yau, there is a unique Ricci-flat Kähler metric \(g\left( t\right) =c_{1}\left( \alpha +t\left[ \pi ^{\ast }\mathcal{L} _{0}\right] \right) \,,\) \(t\in (0,1]\). Then there exists a unique singular Ricci-flat metric \(g_{Y}\) which is a smooth Kähler metric on the non-singular part \(Y_{\mathrm{reg}}\) of \(Y\), see [P. Eyssidieux et al., J. Am. Math. Soc. 22, No. 3, 607–639 (2009; Zbl 1215.32017)]. The main result states that the metric completion of \(\left( Y_{\mathrm{reg}},g_{Y}\right) \) is a metric space with intrinsic metric \(d_{Y} \) homeomorphic to the projective variety \(Y\) itself, that is, the metric completion of \(\left( Y_{\mathrm{reg}},g_{Y}\right) \) is \(\left( Y,d_{Y}\right) \). In addition, the author proves that \(\left( X,g\left( t\right) \right) \) Gromov-Hausdorff converges to \(\left( Y,d_{Y}\right) \) as \(t\rightarrow 0+\). As a corollary (see Corollary 1.2), the author proves a conjecture of P. Candelas and X. de la Ossa [“Comments on conifolds”, Nuclear Phys. B 342 , no. 1, 246–268 (1990)].