  Let $\mathcal{H}$ be a right quaternionic Hilbert space and let $T$ be a
bounded normal right quaternionic linear operator on $\mathcal{H}$. In this
paper, we prove that there exists a unique spectral measure $E$ in
$\mathcal{H}$ such that $$T=\int_{\sigma_S(T)}\lambda dE_\lambda,$$ where
$\sigma_S(T)$ denotes the spherical spectrum of $T$.
