  Molecular helium represents a benchmark system for testing $\textit{ab
initio}$ calculations on few-electron molecules. We report on the determination
of the adiabatic ionization energy of the $a\,^3\Sigma_u^+$ state of He$_2$,
corresponding to the energy interval between the $a\,^3\Sigma_u^+$ ($v''=0$,
$N''=1$) state of He$_2$ and the $X^+\,^2\Sigma_u^+$ ($v^+=0$, $N^+=1$) state
of He${_2}^+$, and of the lowest rotational interval of He${_2}^+$. These
measurements rely on the excitation of metastable He$_2$ molecules to high
Rydberg states using frequency-comb-calibrated continuous-wave UV radiation in
a counter-propagating-laser-beam setup. The observed Rydberg states were
extrapolated to their series limit using multichannel quantum-defect theory.
The ionization energy of He$_2$ ($a\,^3\Sigma_u^+$) and the lowest rotational
interval of He${_2}^+$ ($X^+\,^2\Sigma_u^+$) are 34301.207002(23)$\pm
0.000037_{\mathrm{sys}}$ cm$^{-1}$ and 70.937589(23)$\pm
0.000060_{\mathrm{sys}}$ cm$^{-1}$, respectively.
