  We consider the super-critical contact process on $\mathbb{Z}^d$. It is known
that measures which dominate the upper invariant measure $\mu$ converge
exponentially fast to $\mu$. However, the same is not true for measures which
are below $\mu$, as the time to infect a large empty region is related to its
diameter. The result of this paper is the existence of a spectral gap in
$L^2(\mu)$, that is, the spectrum of the generator is empty inside an open
strip $\{z\in\mathbb{C}: -\lambda<\Im(z)<0\}$ of the complex plane. This is
equivalent to the fact that the variance of the semi-group of the contact
process decays exponentially fast. It is perhaps surprising that the existence
of the spectral gap has not been proven before. One of the reasons is that the
contact process is non-reversible, and hence many methods from spectral theory
are not applicable.
