Magic-angle helical trilayer graphene—three graphene layers sequentially twisted in the same direction by ∼1.8^∘---relaxes into a mosaic of domains that, at zero field, carry opposite valley-resolved Chern numbers, with boundaries hosting a network of gapless conducting modes. Charge transport through this network depends sensitively on how the modes connect and scatter, making well-quantized transport unlikely. Contrary to this expectation, we observe a field-induced Chern gap with Chern number C=-6 emanating from charge neutrality; in this gap, the Hall resistance is quantized to within 2% of the expected value, -h/6e², at 4.6 K. We explain this behavior using both Hofstadter and orbital Zeeman calculations, which show that a moderate magnetic field drives a valley-selective topological transition. Above the transition, the total Chern number of the occupied states in each spin-valley flavor becomes identical across neighboring domains, and the domain-wall modes can become gapped. Though the central valence-band Chern numbers still differ between the two domain types, the observed quantized transport attests to a global gap.