At high baryon chemical potential μ_{B}, the equation of state of QCD allows a weak-coupling expansion in the QCD coupling α_{s}. The result is currently known up to and including the full next-to-next-to-leading order α_{s}^{2}. Starting at this order, the computations are complicated by the modification of particle propagation in a dense medium, which necessitates nonperturbative treatment of the scale α_{s}^{1/2}μ_{B}. We apply a hard-thermal-loop scheme for capturing the contributions of this scale to the weak-coupling expansion, and we use it to determine the leading-logarithm contribution to next-to-next-to-next-to-leading order: α_{s}^{3}ln^{2}α_{s}. This result is the first improvement to the equation of state of massless cold quark matter in 40 years. The new term is negligibly small and thus significantly increases our confidence in the applicability of the weak-coupling expansion.