Removing CO2 from the air with chemicals (Direct Air Capture, DAC) requires a significant amount of energy. Here, we evaluate the cost of co-constructing a solvent DAC process with its energy system. We compare eight energy systems paired with two alternative designs for a liquid-solvent DAC system capturing 1 MtCO2/year, which requires roughly 240 to 300 megawatts of steady power equivalent, 80% thermal and 20% electric. Two energy systems burn natural gas onsite for heat and electricity, capturing nearly all the CO2 released during combustion, and six are all-electric non-fossil systems. The cost of the DAC facility alone contributes $310/tCO2 for a conventional process-based design and $150/tCO2 for a more novel design. When the decomposition of calcium carbonate occurs within a natural-gas-heated calciner, the energy system adds only $80/tCO2 to these costs, assuming $3.25/GJ ($3.43/MMBtu) gas. However, leakage in the natural gas supply chain increases the cost of net capture dramatically: with 2.3% leakage (U.S. national average) and a 20-year Global Warming Potential of 86, costs are about 50% higher. For the all-electric systems, the total capture cost depends on the electricity cost: for each $/MWh of levelized cost of electricity, the total capture cost increases by roughly $2/tCO2. Continuous power is required, because the high-temperature calciner cannot be cycled on and off, so solar and wind power must be supplemented with storage. Our representative capture costs are $250–$440/tCO2 for geothermal energy, $370–$620/tCO2 for nuclear energy (two variants–a light water reactor and small modular nuclear), $360–$570/tCO2 for wind, $430–$690/tCO2 for solar photovoltaics (two variants assuming different daily solar capacities), and $300–$490/tCO2 for a hybrid system with a natural-gas-powered electric calciner.