  We report measurements of partial branching fractions of inclusive charmless
semileptonic $B$-meson decays at the endpoint of the electron momentum
spectrum. The measurements are made in six overlapping momentum intervals that
have lower bounds ranging from 1.9 GeV/$c$ to 2.4 GeV/$c$ and a common upper
bound of 2.6 GeV/$c$, as measured in the centre of mass frame. The results are
based on a sample of 29 million $B\bar{B}$ pairs, accumulated by the Belle
detector at the KEKB asymmetric $e^+e^-$ collider operating on the
$\Upsilon(4S)$ resonance. In the momentum interval ranging from 1.9 GeV/$c$ to
2.6 GeV/$c$ we measure the partial branching fraction $\Delta \mathcal{B}(B\to
X_u e \nu_e)=(8.47 \pm 0.37 \pm 1.53)\times 10^{-4}$, where the first error is
statistical and the second is systematic. A prediction of the partial rate
$R=(21.69 \pm 3.62^{+ 2.18}_{- 1.98}) |V_{ub}|^2 \mathrm{ps}^{-1}$ in this
momentum interval based on theory is calculated with input HQET parameters that
have been derived from Belle's measurement of the $B\to X_s\gamma$ photon
energy spectrum, where the first error is due to the uncertainty on HQET
parameters and the second error is from theory. Using both $\Delta
\mathcal{B}(B\to X_u e \nu_e)$ and $R$ we find $|V_{ub}|=(5.08 \pm 0.47 \pm
0.42^{+ 0.26}_{- 0.23})\times 10^{-3}$, where the first error is from the
partial branching fraction, and the second and third errors are from
uncertainties in $R$.
