The backward region of the H1 detector has been upgraded in order to provide improved measurement of the scattered electron in deep inelastic scattering events. The centerpiece of the upgrade is a high-resolution lead/scintillating-fibre calorimeter. The main design goals of the calorimeter are: good coverage of the region close to the beam pipe, high angular resolution and energy resolution of better than 2% for 30 GeV electrons. The calorimeter should be capable of providing coarse hadronic energy measurement and precise time information to suppress out-of-time background events at the first trigger level. It must be compact due to space restrictions. These requirements were fulfilled by constructing two separate calorimeter sections. The inner electromagnetic section is made of 0.5 mm scintillating plastic fibres embedded in a lead matrix. Its lead-to-fibre ratio is 2.3:1 by volume. The outer hadronic section consists of 1.0 mm diameter fibres with a lead-to-fibre ratio of 3.4:1. The mechanical construction of the new calorimeter and its assembly in the H1 detector are described