The top quark is the heaviest known elementary particle, with a mass about twice the mass of the $W$ and $Z$ bosons of the weak interaction, and about 40 times larger than the mass of its isospin partner, the bottom quark. It decays almost 100% of the time to a $W$ boson and a bottom quark, and the $W$ boson then decays to a lepton and a neutrino or to a quark-antiquark pair. Using top-antitop pairs at the Tevatron proton-antiproton collider, the CDF and {\dzero} collaborations have measured the top quark's mass in different decay channels for integrated luminosities of up to 5.8 fb$^{-1}$. This paper reports on a combination of these measurements that results in a more precise value of the mass than any individual decay channel can provide...