We measure the inclusive semielectronic decay branching fraction of the $D_s^+$ meson. A double-tag technique is applied to $e^+e^-$ annihilation data collected by the BESIII experiment at the BEPCII collider, operating in the center-of-mass energy range $4.178 - 4.230$ GeV. We select positrons from $D_s^+rightarrow Xe^{+} u_e$ with momenta greater than 200 MeV/$c$, and determine the laboratory momentum spectrum, accounting for the effects of detector efficiency and resolution. The total positron yield and semielectronic branching fraction are determined by extrapolating this spectrum below the momentum cutoff. We measure the $D_s^+$ semielectronic branching fraction to be $mathcal{B}left(D_s^+rightarrow Xe^{+} u_eright)=left(6.30pm0.13;(text{stat.})pm 0.10;(text{syst.})right)%$, showing no evidence for unobserved exclusive semielectronic modes. We combine this result with external data taken from literature to determine the ratio of the $D_s^+$ and $D^0$ semielectronic widths, $frac{Gamma(D_{s}^{+}rightarrow Xe^+ u_e)}{Gamma(D^0rightarrow Xe^+ u_e)}=0.790pm 0.016;(text{stat.})pm0.020;(text{syst.})$. Our results are consistent with and more precise than previous measurements.
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