  The young (50-400 Myr) A3V star $\beta$ Leo is a primary target to study the
formation history and evolution of extrasolar planetary systems as one of the
few stars with known hot ($\sim$1600$^\circ$K), warm ($\sim$600$^\circ$K), and
cold ($\sim$120$^\circ$K) dust belt components. In this paper, we present deep
mid-infrared measurements of the warm dust brightness obtained with the Large
Binocular Telescope Interferometer (LBTI) as part of its exozodiacal dust
survey (HOSTS). The measured excess is 0.47\%$\pm$0.050\% within the central
1.5 au, rising to 0.81\%$\pm$0.026\% within 4.5 au, outside the habitable zone
of $\beta$~Leo. This dust level is 50 $\pm$ 10 times greater than in the solar
system's zodiacal cloud. Poynting-Robertson drag on the cold dust detected by
Spitzer and Herschel under-predicts the dust present in the habitable zone of
$\beta$~Leo, suggesting an additional delivery mechanism (e.g.,~comets) or an
additional belt at $\sim$5.5 au. A model of these dust components is provided
which implies the absence of planets more than a few Saturn masses between
$\sim$5 au and the outer belt at $\sim$40 au. We also observationally constrain
giant planets with the LBTI imaging channel at 3.8~$\mu$m wavelength. Assuming
an age of 50 Myr, any planet in the system between approximately 5 au to 50 au
must be less than a few Jupiter masses, consistent with our dust model. Taken
together, these observations showcase the deep contrasts and detection
capabilities attainable by the LBTI for both warm exozodiacal dust and giant
exoplanets in or near the habitable zone of nearby stars.
