  A full renormalisation of the Inert Doublet Model (IDM) is presented and
exploited for a precise calculation of the relic density of Dark Matter (DM) at
one-loop. In this first paper, we study the case of a DM candidate with $m_{\rm
DM} \sim 500$ GeV. In this regime, the co-annihilation channels are important.
We therefore compute, for a wide range of relative velocities, the full
next-to-leading order electroweak corrections to 7 annihilation/co-annihilation
processes that contribute $\sim$ 70\% to the relic density of DM. These
corrected cross-sections are interfaced with {\tt micrOMEGAs} to obtain the
one-loop correction to the freeze-out relic density. Due to the accurate
measurement of this observable, the one-loop corrections are relevant. We
discuss the one-loop renormalisation scheme dependence and point out the
influence, at one-loop, of a parameter that solely describes the scattering in
the dark sector. A tree-level computation of the relic density is not sensitive
to this parameter.
