  There are very few confirmed black holes with a mass that could be $\sim\!
4\, M_\odot$ and no neutron stars with masses greater than $\sim\! 2\,
M_\odot$, creating a gap in the observed distribution of compact star masses.
Some black holes with masses between 2 and $4\, M_\odot$ might be hiding among
other X-ray sources, whose masses are difficult to measure. We present new
high-speed optical photometry of the low-mass X-ray binary V1408 Aql (= 4U
1957+115), which is a persistent X-ray source thought to contain a black hole.
The optical light curve of V1408~Aql shows a nearly sinusoidal modulation at
the orbital period of the system superimposed on large night-to-night
variations in mean intensity. We combined the new photometry with
previously-published photometry to derive a more precise orbital period, $P =
0.388893(3)$\ d, and to better define the orbital light curve and
night-to-night variations. The orbital light curve agrees well with a model in
which the modulation is caused entirely by the changing aspect of the heated
face of the secondary star. The lack of eclipses rules out orbital inclinations
greater than $65^{\circ}$. Our best models for the orbital light curve favor
inclinations near $13^{\circ}$ and black hole masses near $3\, M_\odot$ with a
90\% upper bound of $6.2\, M_\odot$, and a lower bound of $2.0\, M_\odot$
imposed solely by the maximum mass of neutron stars. We favor a black hole
primary over a neutron star primary based on evidence from the X-ray spectra,
the high spin of the compact object, and the fact that a type I X-ray burst has
not been observed for this system. Although uncertainties in the data and the
models allow higher masses, possibly much higher masses, the compact star in
V1408~Aql is a viable candidate for a black hole lying in the mass gap.
