  We present the first high-resolution, simultaneous observations of the solar
chromosphere in the optical and millimeter wavelength ranges, obtained with
ALMA and the IBIS instrument at the Dunn Solar Telescope. In this paper we
concentrate on the comparison between the brightness temperature observed in
ALMA Band 3 (3 mm; 100 GHz) and the core width of the H$\alpha$ 656.3 nm line,
previously identified as a possible diagnostic of the chromospheric
temperature. We find that in the area of plage, network and fibrils covered by
our FOV the two diagnostics are well correlated, with similar spatial
structures observed in both. The strength of the correlation is remarkable,
given that the source function of the mm-radiation obeys local thermodynamic
equilibrium, while the H$\alpha$ line has a source function that deviates
significantly from the local Planck function. The observed range of ALMA
brightness temperatures is sensibly smaller than the temperature range that was
previously invoked to explain the observed width variations in H$\alpha$. We
employ analysis from forward modeling with the RH code to argue that the strong
correlation between H$\alpha$ width and ALMA brightness temperature is caused
by their shared dependence on the population number $n_2$ of the first excited
level of hydrogen. This population number drives millimeter opacity through
hydrogen ionization via the Balmer continuum, and H$\alpha$ width through a
curve-of-growth-like opacity effect. Ultimately, the $n_2$ population is
regulated by the enhancement or lack of downward Ly$\alpha$ flux, which
coherently shifts the formation height of both diagnostics to regions with
different temperature, respectively.
