We prove a differential analogue of Hilbert's irreducibility theorem. Let $\mathcal{L}$ be a linear differential operator with coefficients in $C(\mathcal{X})(x)$ that is irreducible over $\overline{C(\mathcal{X})}(x)$, where $\mathcal{X}$ is an irreducible affine algebraic variety over an algebraically closed field $C$ of characteristic zero. We show that the set of $c\in \mathcal{X}(C)$ such that the specialized operator ${\mathcal{L}}^c$ of $\mathcal{L}$ remains irreducible over $C(x)$ is Zariski dense in $\mathcal{X}(C)$.