In search of the origin of superconductivity in diluted rhenium superconductors and their significantly enhanced $T_c$ compared to pure Be (0.026 K), we investigated the intermetallic ReBe$_{22}$ compound, mostly by means of muon-spin rotation/relaxation ($mu$SR). At a macroscopic level, its bulk superconductivity (with $T_c=9.4$ K) was studied via electrical resistivity, magnetization, and heat-capacity measurements. The superfluid density, as determined from transverse-field $mu$SR and electronic specific-heat measurements, suggest that ReBe$_{22}$ is a fully-gapped superconductor with some multigap features. The larger gap value, $Delta_0^l=1.78$ k$_mathrm{B}T_c$, with a weight of almost 90%, is slightly higher than that expected from the BCS theory in the weak-coupling case. The multigap feature, rather unusal for an almost elemental superconductor, is further supported by the field-dependent specific-heat coefficient, the temperature dependence of the upper critical field, as well as by electronic band-structure calculations. The absence of spontaneous magnetic fields below $T_c$, as determined from zero-field $mu$SR measurements, indicates a preserved time-reversal symmetry in the superconducting state of ReBe$_{22}$. In general, we find that a dramatic increase in the density of states at the Fermi level and an increase in the electron-phonon coupling strength, both contribute to the highly enhanced $T_c$ value of ReBe$_{22}$.
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