The release of fossil fuel CO2 to the atmosphere by human activity has been implicated as the predominant cause of recent global climate change. The ocean plays a crucial role in mitigating the effects of this perturbation to the climate system, sequestering 20 to 35 per cent of anthropogenic CO2 emissions. Although much progress has been made in recent years in understanding and quantifying this sink, considerable uncertainties remain as to the distribution of anthropogenic CO2 in the ocean, its rate of uptake over the industrial era, and the relative roles of the ocean and terrestrial biosphere in anthropogenic CO2 sequestration. Here we address these questions by presenting an observationally based reconstruction of the spatially resolved, time-dependent history of anthropogenic carbon in the ocean over the industrial era. Our approach is based on the recognition that the transport of tracers in the ocean can be described by a Green's function, which we estimate from tracer data using a maximum entropy deconvolution technique. Our results indicate that ocean uptake of anthropogenic CO2 has increased sharply since the 1950s, with a small decline in the rate of increase in the last few decades. We estimate the inventory and uptake rate of anthropogenic CO2 in 2008 at 140+/-25PgC and 2.3+/-0.6PgCyr-1, respectively. We find that the Southern Ocean is the primary conduit by which this CO2 enters the ocean (contributing over 40 per cent of the anthropogenic CO2 inventory in the ocean in 2008). Our results also suggest that the terrestrial biosphere was a source of CO2 until the 1940s, subsequently turning into a sink. Taken over the entire industrial period, and accounting for uncertainties, we estimate that the terrestrial biosphere has been anywhere from neutral to a net source of CO2, contributing up to half as much CO2 as has been taken up by the ocean over the same period.