The rate of global-mean sea-level rise since 1900 has varied over time, but the contributing factors are still poorly understood(1). Previous assessments found that the summed contributions of ice-mass loss, terrestrial water storage and thermal expansion of the ocean could not be reconciled with observed changes in global-mean sea level, implying that changes in sea level or some contributions to those changes were poorly constrained(2,3). Recent improvements to observational data, our understanding of the main contributing processes to sea-level change and methods for estimating the individual contributions, mean another attempt at reconciliation is warranted. Here we present a probabilistic framework to reconstruct sea level since 1900 using independent observations and their inherent uncertainties. The sum of the contributions to sea-level change from thermal expansion of the ocean, ice-mass loss and changes in terrestrial water storage is consistent with the trends and multidecadal variability in observed sea level on both global and basin scales, which we reconstruct from tide-gauge records. Ice-mass loss-predominantly from glaciers-has caused twice as much sea-level rise since 1900 as has thermal expansion. Mass loss from glaciers and the Greenland Ice Sheet explains the high rates of global sea-level rise during the 1940s, while a sharp increase in water impoundment by artificial reservoirs is the main cause of the lower-than-average rates during the 1970s. The acceleration in sea-level rise since the 1970s is caused by the combination of thermal expansion of the ocean and increased ice-mass loss from Greenland. Our results reconcile the magnitude of observed global-mean sea-level rise since 1900 with estimates based on the underlying processes, implying that no additional processes are required to explain the observed changes in sea level since 1900.