The heat capacities of water and NaCl(aq) from 0.05 mol(.)kg(-1) to 6 mol(.)kg(-1) were measured from 285 K to a lower temperature of 202 K < T < 236 K, at 0.1 MPa, dependent on composition. These measurements were performed with a differential scanning calorimeter (DSC). A cooling scan method permitted supercooling of water and aqueous solutions well below their normal freezing points. Calibration methods for the DSC and measurement methods for the heat capacity using cooling scans are described. The uncertainties of temperature and heat-flow calibration coefficient for the DSC were +/- 0.08 K and 10.3%. respectively. The heat capacity of dilute NaCl(aq) increases with decreasing temperature below 270 K, qualitatively similar to the effect observed for water. At larger concentrations of NaCl(aq), this behavior reverses and the heat capacity only decreases with decreasing temperature. An equation of state for the NaCl + H2O system was generated on the basis of the present measurements as well as other experimental results that spanned the range of temperature of approximately 250-600 K and, where available, the range of pressure from the vapor pressure of the solution to 100 MPa. The present results for NaCl(aq) were examined in the context of the three competing interpretations of the anomalous properties of supercooled water. These are the stability limit conjecture, the critical point scenario, and the singularity-free interpretation. The present results lend support to the singularity-free interpretation. They also indicate that if a second critical point for water does exist at low temperature and high pressure, then the critical line for NaCl(aq) moves to lower temperatures with increasing concentration of NaCl. Solute activity coefficients from the equation of state for NaCl + H2O were also examined for evidence of ion association in the solution at supercooled temperatures.