We investigate the Gilbert damping parameter α for rare earth (RE)-transition metal (TM) ferrimagnets over a wide temperature range. Extracted from the field-driven magnetic domain-wall mobility, α was as low as the order of 10^{-3} and was almost constant across the angular momentum compensation temperature T_{A}, starkly contrasting previous predictions that α should diverge at T_{A} due to a vanishing total angular momentum. Thus, magnetic damping of RE-TM ferrimagnets is not related to the total angular momentum but is dominated by electron scattering at the Fermi level where the TM has a dominant damping role. This low value of the Gilbert damping parameter suggests that ferrimagnets can serve as versatile platforms for low-dissipation high-speed magnetic devices.