The liquid argon calorimeter of the H 1 detector presently taking data at the HERA ep collider at DESY, Hamburg, is described. The main physics requirements and the most salient design features relevant to this calorimeter are given. The aim to have smooth and hermetic calorimetric coverage over the polar angular range 4° ≤ θ ≤ 154° is achieved by a single liquid argon cryostat containing calorimeter stacks structured in wheels and octants for easy handling. The absorber materials used are lead in the electromagnetic part and stainless steel in the hadronic part. The read-out system is pipelined to reduce the dead time induced by the high trigger rate expected at the HERA collider where consecutive bunches are separated in time by 96 ns. The main elements of the calorimeter, such as the cryostat, with its associated cryogenics, the stack modules, the read-out, calibration and trigger electronics as well as the data acquisition system are described. Performance results from data taken in calibration runs with full size H 1 calorimeter stacks at a CERN test beam, as well as results from data collected with the complete H 1 detector using cosmic rays during the initial phase of ep operations are presented. The observed energy resolutions and linearities are well in agreement with the requirements