Advanced beyond-silicon electronic technology requires both channel materials and also ultralow-resistance contacts to be discovered1,2. Atomically thin two-dimensional semiconductors have great potential for realizing high-performance electronic devices1,3. However, owing to metal-induced gap states (MIGS)4–7, energy barriers at the metal–semiconductor interface—which fundamentally lead to high contact resistance and poor current-delivery capability—have constrained the improvement of two-dimensional semiconductor transistors so far2,8,9. Here we report ohmic contact between semimetallic bismuth and semiconducting monolayer transition metal dichalcogenides (TMDs) where the MIGS are sufficiently suppressed and degenerate states in the TMD are spontaneously formed in contact with bismuth. Through this approach, we achieve zero Schottky barrier height, a contact resistance of 123 ohm micrometres and an on-state current density of 1,135 microamps per micrometre on monolayer MoS2; these two values are, to the best of our knowledge, the lowest and highest yet recorded, respectively. We also demonstrate that excellent ohmic contacts can be formed on various monolayer semiconductors, including MoS2, WS2 and WSe2. Our reported contact resistances are a substantial improvement for two-dimensional semiconductors, and approach the quantum limit. This technology unveils the potential of high-performance monolayer transistors that are on par with state-of-the-art three-dimensional semiconductors, enabling further device downscaling and extending Moore’s law.