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 MtCO(2)/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/tCO(2) for a conventional process-based design and $150/tCO(2) for a more novel design. When the decomposition of calcium carbonate occurs within a natural-gas-heated calciner, the energy system adds only $80/tCO(2) 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/tCO(2). 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/tCO(2) for geothermal energy, $370-$620/tCO(2) for nuclear energy (two variants-a light water reactor and small modular nuclear), $360-$570/tCO(2) for wind, $430-$690/tCO(2) for solar photovoltaics (two variants assuming different daily solar capacities), and $300-$490/tCO(2) for a hybrid system with a natural-gas-powered electric calciner.