Measurement of the Inclusive Semi-electronic $D^0$ Branching Fraction

Using the angular correlation between the $\pi^+$ emitted in a $D^{*+} \rightarrow D^0 \pi^+$ decay and the $e^+$ emitted in the subsequent $D^0 \rightarrow Xe^+\nu$ decay, we have measured the branching fraction for the inclusive semi-electronic decay of the $D^0$ meson to be: {\cal B}(D^0 \rightarrow X e^+ \nu) = [6.64 \pm 0.18 (stat.) \pm 0.29 (syst.)] \%. The result is based on 1.7 fb$^{-1}$ of $e^+e^-$ collisions recorded by the CLEO II detector located at the Cornell Electron Storage Ring (CESR). Combining the analysis presented in this paper with previous CLEO results we find, \frac{{\cal B} (D^0 \rightarrow X e^+ \nu)} {{\cal B} (D^0 \rightarrow K^- \pi^+)} = 1.684 \pm 0.056 (stat.) \pm 0.093(syst.) and \frac{{\cal B}(D\rightarrow K^-e^+\nu)} {{\cal B}(D\rightarrow Xe^+\nu)} = 0.581 \pm 0.023 (stat.) \pm 0.028(syst.). The difference between the inclusive rate and the sum of the measured exclusive branching fractions (measured at CLEO and other experiments) is $(3.3 \pm 7.2) \%$ of the inclusive rate.