  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
$D_s^+\rightarrow Xe^{+}\nu_e$ 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
$\mathcal{B}\left(D_s^+\rightarrow
Xe^{+}\nu_e\right)=\left(6.30\pm0.13\;(\text{stat.})\pm
0.10\;(\text{syst.})\right)\%$, 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, $\frac{\Gamma(D_{s}^{+}\rightarrow
  Xe^+\nu_e)}{\Gamma(D^0\rightarrow Xe^+\nu_e)}=0.790\pm
0.016\;(\text{stat.})\pm0.020\;(\text{syst.})$. Our results are consistent with
and more precise than previous measurements.
