Ice cores contain a record of snow precipitation that includes information about past atmospheric circulation and mass imbalance in the polar regions. We present a novel approach to reconstruct a climatic record - by both optimally dating an ice core and deriving from it a detailed accumulation history - that uses an adjoint-based method. The motivation of our work is the recent application of phase-sensitive radar which measures the vertical velocity of an ice column. The velocity is dependent on the history of subsequent snow accumulation, compaction and compression; in our inverse formulation of this problem, measured vertical velocity profiles can be utilized directly, thereby reducing the uncertainty introduced by ice-flow modelling. We first apply our method to synthetic data in order to study its capability and the effect of noise and gaps in the age-depth observations. The method is then applied to the ice core retrieved from James Ross Island, Antarctica. We show that the method is robust and that the results depend on the quality of the age-depth observations and the derived flow regime around the core site. The method facilitates the incorporation of increasing detail provided by ice-core analysis together with observed full-depth velocity in order to construct a complete climatic record of the polar regions.