Abstract: A method and apparatus for the hemodialysis of the blood of kidney failure patients. The concentration of the dialysis solution is initially selected so that the solution osmolality approximates the osmolality of the patient's blood. The dialysis solution concentration is then reduced linearly as a function of time as the dialysis is carried out. The conductivity of the dialysis solution is measured and compared with high and low limit references. If the conductivity is outside the limits, an alarm is given and the concentration of the dialysis solution is held constant. A timer controls a proportional pumping means to reduce the concentration of the dialysis solution and concurrently reduces the conductivity reference limits for the alarm.