  We report results from very long baseline interferometric (VLBI) observations
of the supermassive black hole in the Galactic center, Sgr A*, at 1.3 mm (230
GHz). The observations were performed in 2013 March using six VLBI stations in
Hawaii, California, Arizona, and Chile. Compared to earlier observations, the
addition of the APEX telescope in Chile almost doubles the longest baseline
length in the array, provides additional {\it uv} coverage in the N-S
direction, and leads to a spatial resolution of $\sim$30 $\mu$as ($\sim$3
Schwarzschild radii) for Sgr A*. The source is detected even at the longest
baselines with visibility amplitudes of $\sim$4-13% of the total flux density.
We argue that such flux densities cannot result from interstellar refractive
scattering alone, but indicate the presence of compact intrinsic source
structure on scales of $\sim$3 Schwarzschild radii. The measured nonzero
closure phases rule out point-symmetric emission. We discuss our results in the
context of simple geometric models that capture the basic characteristics and
brightness distributions of disk- and jet-dominated models and show that both
can reproduce the observed data. Common to these models are the brightness
asymmetry, the orientation, and characteristic sizes, which are comparable to
the expected size of the black hole shadow. Future 1.3 mm VLBI observations
with an expanded array and better sensitivity will allow a more detailed
imaging of the horizon-scale structure and bear the potential for a deep
insight into the physical processes at the black hole boundary.
