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.
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