  We report the formation of intermediate-mass black holes (IMBHs) in suites of
numerical $N$-body simulations of Population III remnant black holes (BHs)
embedded in gas-rich protogalaxies at redshifts $z\gtrsim10$. We model the
effects of gas drag on the BHs' orbits, and allow BHs to grow via gas
accretion, including a mode of hyper-Eddington accretion in which photon
trapping and rapid gas inflow suppress any negative radiative feedback. Most
initial BH configurations lead to the formation of one (but never more than
one) IMBH in the center of the protogalaxy, reaching a mass of
$10^{3-5}\mathrm{M}_{\odot}$ through hyper-Eddington growth. Our results
suggest a viable pathway to forming the earliest massive BHs in the centers of
early galaxies. We also find that the nuclear IMBH typically captures a
stellar-mass BH companion, making these systems observable in gravitational
waves as extreme mass-ratio inspirals (EMRIs) with \textit{eLISA}.
