  Draco C1 is a known symbiotic binary star system composed of a carbon red
giant and a hot, compact companion -- likely a white dwarf -- belonging to the
Draco dwarf spheroidal galaxy. From near-infrared spectroscopic observations
taken by the Apache Point Observatory Galactic Evolution Experiment (APOGEE-2),
part of Sloan Digital Sky Survey IV, we provide updated stellar parameters for
the cool, giant component, and constrain the temperature and mass of the hot,
compact companion. Prior measurements of the periodicity of the system, based
on only a few epochs of radial velocity data or relatively short baseline
photometric observations, were sufficient only to place lower limits on the
orbital period ($P > 300$ days). For the first time, we report precise orbital
parameters for the binary system: With 43 radial velocity measurements from
APOGEE spanning an observational baseline of more than 3 years, we definitively
derive the period of the system to be $1220.0^{+3.7}_{-3.5}$ days. Based on the
newly derived orbital period and separation of the system, together with
estimates of the radius of the red giant star, we find that the hot companion
must be accreting matter from the dense wind of its evolved companion.
