  The Galactic Center black hole Sagittarius A* (Sgr A*) is a prime observing
target for the Event Horizon Telescope (EHT), which can resolve the 1.3 mm
emission from this source on angular scales comparable to that of the general
relativistic shadow. Previous EHT observations have used visibility amplitudes
to infer the morphology of the millimeter-wavelength emission. Potentially much
richer source information is contained in the phases. We report on 1.3 mm phase
information on Sgr A* obtained with the EHT on a total of 13 observing nights
over 4 years. Closure phases, the sum of visibility phases along a closed
triangle of interferometer baselines, are used because they are robust against
phase corruptions introduced by instrumentation and the rapidly variable
atmosphere. The median closure phase on a triangle including telescopes in
California, Hawaii, and Arizona is nonzero. This result conclusively
demonstrates that the millimeter emission is asymmetric on scales of a few
Schwarzschild radii and can be used to break 180-degree rotational ambiguities
inherent from amplitude data alone. The stability of the sign of the closure
phase over most observing nights indicates persistent asymmetry in the image of
Sgr A* that is not obscured by refraction due to interstellar electrons along
the line of sight.
