Long memory is a hydrological property that can lead to prolonged droughts or the temporal clustering of extreme floods in a river. Analyses of 28 long (up to 145 years), continuous instrumental runoff series from six European, American, and African rivers reveal that this effect increases downstream. Simulations reproduce the increase qualitatively and show that a river network aggregates short-memory precipitation and converts it into long-memory runoff. In view of projected changes in climate and the hydrological cycle, these findings show that decadal-scale variations in drought or flood risk can be predicted for individual rivers, with higher predictability downstream. Spatial aggregation may also explain the emergence of long memory in other networks, such as the brain or those formed by computers.