At high baryon chemical potential mu_B, the equation of state of QCD allows a weak-coupling expansion in the QCD coupling alpha_s. The result is currently known up to and including the full next-to-next-to-leading order (NNLO) alpha_s^2. Starting at this order, the computations are complicated by the modification of particle propagation in a dense medium, which necessitates non-perturbative treatment of the scale Sqrt(alpha_s)*mu_B. In this work, we apply a Hard-Thermal-Loop scheme for capturing the contributions of this scale to the weak-coupling expansion, and use it to determine the leading-logarithm contribution to NNNLO: alpha_s^3 (ln alpha_s)^2. 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.