The satellite galaxies of the Milky Way (MW) are effective probes of the underlying dark matter (DM) substructure, which is sensitive to the nature of the DM particle. In particular, a class of DM models have a power spectrum cut-off on the mass scale of dwarf galaxies and thus predict only small numbers of substructures below the cut-off mass. This makes the MW satellite system appealing to constrain the DM properties: feasible models must produce enough substructure to host the number of observed Galactic satellites. Here, we compare theoretical predictions of the abundance of DM substructure in thermal relic warm DM (WDM) models with estimates of the total satellite population of the MW. This produces conservative robust lower limits on the allowed mass, $m_mathrm{th}$, of the thermal relic WDM particle. As the abundance of satellite galaxies depends on the MW halo mass, we marginalize over the corresponding uncertainties and rule out $m_mathrm{th} leq 2.02, mathrm{keV}$ at 95 per cent confidence independently of assumptions about galaxy formation processes. Modelling some of these - in particular, the effect of reionization, which suppresses the formation of dwarf galaxies - strengthens our constraints on the DM properties and excludes models with $m_mathrm{th} leq 3.99, mathrm{keV}$ in our fiducial model. We also find that thermal relic models cannot produce enough satellites if the MW halo mass is $M_{200}leq 0.6times 10^{12}, mathrm{M_odot}$, which imposes a lower limit on the MW halo mass in CDM. We address several observational and theoretical uncertainties and discuss how improvements in these will strengthen the DM mass constraints.
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