On a resolution of singularities with two strata

Let $X$ be a complex, irreducible, quasi-projective variety, and $\pi:\widetilde X\to X$ a resolution of singularities of $X$. Assume that the singular locus ${\text{Sing}}(X)$ of $X$ is smooth, that the induced map $\pi^{-1}({\text{Sing}}(X))\to {\text{Sing}}(X)$ is a smooth fibration admitting a cohomology extension of the fiber, and that $\pi^{-1}({\text{Sing}}(X))$ has a negative normal bundle in $\widetilde X$. We present a very short and explicit proof of the Decomposition Theorem for $\pi$, providing a way to compute the intersection cohomology of $X$ by means of the cohomology of $\widetilde X$ and of $\pi^{-1}({\text{Sing}}(X))$. Our result applies to special Schubert varieties with two strata, even if $\pi$ is non-small. And to certain hypersurfaces of $\mathbb P^5$ with one-dimensional singular locus.