In a recent Letter [T.~Dornheim \textit{et al.}, Phys. Rev. Lett.
\textbf{117}, 156403 (2016)], we presented the first \textit{ab initio} quantum
Monte-Carlo (QMC) results of the warm dense electron gas in the thermodynamic
limit. However, a complete parametrization of the exchange-correlation free
energy with respect to density, temperature, and spin polarization remained out
of reach due to the absence of (i) accurate QMC results below
$\theta=k_\text{B}T/E_\text{F}=0.5$ and (ii) of QMC results for spin
polarizations different from the paramagnetic case. Here we overcome both
remaining limitations. By closing the gap to the ground state and by performing
extensive QMC simulations for different spin polarizations, we are able to
obtain the first complete \textit{ab initio} exchange-correlation free energy
functional; the accuracy achieved is an unprecedented $\sim 0.3\%$. This also
allows us to quantify the accuracy and systematic errors of various previous
approximate functionals.