  We present a measurement of the $\nu_e$-interaction rate in the MicroBooNE
detector that addresses the observed MiniBooNE anomalous low-energy excess
(LEE). The approach taken isolates neutrino interactions consistent with the
kinematics of charged-current quasi-elastic (CCQE) events. The topology of such
signal events has a final state with 1 electron, 1 proton, and 0 mesons
($1e1p$). Multiple novel techniques are employed to identify a $1e1p$ final
state, including particle identification that use two methods of
deep-learning-based image identification, and event isolation using a boosted
decision-tree ensemble trained to recognize two-body scattering kinematics.
This analysis selects 25 $\nu_e$-candidate events in the reconstructed neutrino
energy range of 200--1200\,MeV, while $29.0 \pm 1.9_\text{(sys)} \pm
5.4_\text{(stat)}$ are predicted when using $\nu_\mu$ CCQE interactions as a
constraint. We use a simplified model to translate the MiniBooNE LEE
observation into a prediction for a $\nu_e$ signal in MicroBooNE. A $\Delta
\chi^2$ test statistic, based on the combined Neyman--Pearson $\chi^2$
formalism, is used to define frequentist confidence intervals for the LEE
signal strength. Using this technique, in the case of no LEE signal, we expect
this analysis to exclude a normalization factor of 0.75 (0.98) times the median
MiniBooNE LEE signal strength at 90\% ($2\sigma$) confidence level, while the
MicroBooNE data yield an exclusion of 0.25 (0.38) times the median MiniBooNE
LEE signal strength at 90\% ($2\sigma$) confidence
