  We report on the first millisecond timescale radio interferometric search for
the new class of transient known as fast radio bursts (FRBs). We used the Very
Large Array (VLA) for a 166-hour, millisecond imaging campaign to detect and
precisely localize an FRB. We observed at 1.4 GHz and produced visibilities
with 5 ms time resolution over 256 MHz of bandwidth. Dedispersed images were
searched for transients with dispersion measures from 0 to 3000 pc/cm3. No
transients were detected in observations of high Galactic latitude fields taken
from September 2013 though October 2014. Observations of a known pulsar show
that images typically had a thermal-noise limited sensitivity of 120 mJy/beam
(8 sigma; Stokes I) in 5 ms and could detect and localize transients over a
wide field of view. Our nondetection limits the FRB rate to less than
7e4/sky/day (95% confidence) above a fluence limit of 1.2 Jy-ms. Assuming a
Euclidean flux distribution, the VLA rate limit is inconsistent with the
published rate of Thornton et al. We recalculate previously published rates
with a homogeneous consideration of the effects of primary beam attenuation,
dispersion, pulse width, and sky brightness. This revises the FRB rate downward
and shows that the VLA observations had a roughly 60% chance of detecting a
typical FRB and that a 95% confidence constraint would require roughly 500
hours of similar VLA observing. Our survey also limits the repetition rate of
an FRB to 2 times less than any known repeating millisecond radio transient.
