  We consider the damped Schr\"odinger semigroup $e^{-it \frac{d^2}{dx^2}}$ on
the tadpole graph ${\mathcal R}$. We first give a careful spectral analysis and
an appropriate decomposition of the kernel of the resolvent. As a consequence
and by showing that the generalized eigenfunctions form a Riesz basis of some
subspace of $L^2({\mathcal R})$, we prove that the corresponding energy decay
exponentially.
