A HDR closed loop geothermal steam generating system comprises of three large vertical bores driven in solid rock to a depth of 2 to 3 miles and terminating in an enlarged bottom chamber from which a series of horizontal tunnels radiate. A plurality of small bores are then driven down from each horizontal tunnel to a depth where the rock temperature is high enough for steam generation. A steam generating chamber is placed at the foot of each small bore to which feed water is supplied. Steam is piped from the steam generating chambers to a common header in each horizontal tunnel. Each header feeds into a steam riser in one of the large vertical bores which conveys the steam to the surface. Steam condensate is recycled as feed water without the need of pumps.