Anionic DNA and cationic surfactants form charge neutral complexes that contain finite amounts of water. There is a strong electrostatic attraction between the oppositely charged species, and the finite swelling is caused by an opposing repulsive force. Adding NaCl to the complexes provides an opportunity to modulate the strength of the electrostatic attraction. The thermodynamics of the isothermal swelling process has been experimentally characterized using a calorimetric technique monitoring both the free energy and the enthalpy. The experimental results are quantitatively analyzed in calculations using the Poisson-Boltzmann equation to describe the electrostatic effects. The main findings are as follows: (i) Addition of salt results in an increased swelling at a given water activity. (ii) The effect of the salt can be quantitatively modeled on the basis of the Poisson-Boltzmann equation with a dielectric description of the water. (iii) There exists a short-range repulsive force between DNA double helices and surfactant aggregates. (iv) Solid NaCl dissolves in the complex at water activities in the range 0.5-0.6 rather than at 0.74 as in a saturated aqueous solution. (v) The heat of solution of NaCl in the complexes is around +1.6 +/- 0.5 kJ/mol, surprisingly close to the values found for the dissolution into bulk aqueous solutions.