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Ice sheets in Greenland and Antarctica are based on continental rock and are coupled strongly to changes in global sea level. Their beds may be frozen or thawed, but it has not been known how much or where basal melt occurs, except in special cases where a borehole has reached the bed or where airborne radar has revealed subglacial lakes and reflective ice stream beds in Antarctica. We have used a previously published technique to detect subglacial melt water in the general case and have here applied it to radar-sounder data collected over northern and central Greenland. We have found extensive subglacial water along between 13% and 20% of the flight paths. This paper provides maps of the measured locations and probable extent of subglacial water.