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We study the Hilbert scheme ${\mathrm{Hilb}}^{sc}V$ of smooth connected curves on a smooth del Pezzo $3$-fold $V$. We prove that any degenerate curve $C$, any curve $C$ contained in a smooth hyperplane section $S$ of $V$, does not deform to a non-degenerate curve if the following two conditions are satisfied: (i) $\chi (V,{ℐ}_C\left(S\right))\ge 1$ and (ii) for every line $\ell$ on $S$ such that $\ell \cap C=\varnothing$, the normal bundle $N_{\ell /V}$ is trivial (  $N_{\ell /V}\simeq {{𝒪}_{{\mathbb{P}}^1}}^{\oplus 2}$). As a consequence, we prove an analogue (for ${\mathrm{Hilb}}^{sc}V$) of a conjecture of J. O. Kleppe, which is concerned with non-reduced components of the...