  We determine the pressure of a cold and dense electron gas to a nearly
complete next-to-next-to-next-to-leading order (N$^3$LO) in the fine-structure
constant $\alpha_e$, utilizing a new result for the two-loop photon self-energy
from a companion paper. Our result contains all infrared-sensitive
contributions to the pressure at this order, including the coefficient of the
$O(\alpha_e^3 \ln \alpha_e^{ })$ term, and leaves only a single coefficient
associated with the contributions of unresummed hard momenta undetermined.
Moreover, we explicitly demonstrate the complete cancellation of infrared
divergences according to the effective field theory paradigm by determining
part of the hard contributions at this order. Our calculation provides the
first improvement to a 45-year-old milestone result and demonstrates the
feasibility of the corresponding N$^3$LO calculation for cold and dense quark
matter.
