In search of the origin of superconductivity (SC) in diluted rhenium superconductors and their significantly enhanced T_c compared to pure Be (0.026 K), we investigated the intermetallic ReBe₂₂ compound, mostly by means of muon-spin rotation/relaxation (μSR). At a macroscopic level, its bulk SC (with T_c = 9.4 K) was studied via electrical resistivity, magnetization, and heat-capacity measurements. The superfluid density, as determined from transverse-field μSR and electronic specific-heat measurements, suggest that ReBe₂₂ is a fully-gapped superconductor with some multigap features. The larger gap value, Δ^l₀ = 1.78 k_BT_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 unusual 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 μSR measurements, indicates a preserved time-reversal symmetry in the superconducting state of ReBe₂₂. 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 ReBe₂₂.