  We present the first observational constraint on the Br-gamma recombination
line emission associated with the supermassive black hole at the center of our
Galaxy, known as Sgr A*. By combining 13 years of data with the Adaptive Optics
fed integral field spectrograph OSIRIS at the W. M. Keck Observatory obtained
as part of the Galactic Center Orbits Initiative, we extract the near-infrared
spectrum within ~0.2'' of the black hole and we derive an upper limit on the
Br-gamma flux. The aperture was set to match the size of the disk-like
structure that was recently reported based on millimeter-wave ALMA observations
of the hydrogen recombination line, H30-alpha. Our stringent upper limit is at
least a factor of 80 (and up to a factor of 245) below what would be expected
from the ALMA measurements and strongly constrains possible interpretation of
emission from this highly under-luminous supermassive black hole.
