A new flow conductance apparatus for measurements of aqueous solutions near the critical point of water has been used to measure the conductance of aqueous LiCl, NaCl, NaBr, and CsBr at pressures up to 28 MPa and temperatures between 579 and 673 K (water densities from 700 to 260 kg m(-3)). At a water density of 300 kg m(-3) the new instrument can make measurements with a precision of about 1% at concentrations as low as 2 x 10(-7) mol dm(-3). At higher concentrations and higher water densities the precision is 0.1% or better, Values for infinite dilution conductance Lambda(0) and dissociation constant K-m(theta) for the four salts under these conditions are reported. The differences in Lambda(0) and K-m(theta), for the four salts at the same temperature and pressure are quite small. This instrument extends the region where accurate measurements of Lambda(0) and K-m(theta) can be made. An analysis of the behavior of Lambda(0) indicates that on average only a small number of water molecules are carried along by the ion even under conditions where there is extensive long range hydration, N-ex approximate to 150.