Summary: We prove the existence of a maximum for the first Steklov-Dirichlet eigenvalue in the class of convex sets with a fixed spherical hole, under volume constraint. More precisely, if \(\Omega=\Omega_0 \setminus\overline{B}_{R_1} \), where \(B_{R_1}\) is the ball centered at the origin with radius \(R_1>0\) and \(\Omega_0\subset\mathbb{R}^n\), \(n\geq 2\), is an open, bounded and convex set such that \(B_{R_1}\Subset \Omega_0\), then the first Steklov-Dirichlet eigenvalue \(\sigma_1(\Omega)\) has a maximum when \(R_1\) and the measure of \(\Omega\) are fixed. Moreover, if \(\Omega_0\) is contained in a suitable ball, we prove that the spherical shell is the maximum.