We describe a new balloon-borne instrument (PoGOLite) capable of detecting 10% polarisation from 200 mCrab point-like sources between 25 keV and 80 keV in one 6 hour flight. Polarisation measurements in the soft gamma-ray band are expected to provide a powerful probe into high-energy emission mechanisms as well as the distribution of magnetic fields, radiation fields and interstellar matter. Synchrotron radiation, inverse Compton scattering and propagation through high magnetic fields are likely to produce high degrees of polarisation in the energy band of the instrument. We demonstrate, through tests at accelerators, with radioactive sources and through computer simulations, that PoGOLite will be able to detect degrees of polarisation as predicted by models for several classes of high-energy sources. At present, only exploratory polarisation measurements have been carried out in the soft gamma-ray band. Reduction of the large background produced by cosmic-ray particles while securing a large effective area has been the greatest challenge. PoGOLite uses Compton scattering and photo-absorption in an array of 217 well-type phoswich detector cells made of plastic and BGO scintillators surrounded by a BGO anticoincidence shield and a thick polyethylene neutron shield. The narrow field of view (FWHM = 1.25 msr, 2.0 degrees × 2.0 degrees) obtained with detector cells and the use of thick background shields warrant a large effective area for polarisation measurements (∼228 cm 2 at E = 40 keV) without sacrificing the signal-to-noise ratio. Simulation studies for an atmospheric overburden of 3-4 g/cm 2 indicate that neutrons and gamma-rays entering the PDC assembly through the shields are dominant backgrounds. Off-line event selection based on recorded phototube waveforms and Compton kinematics reduce the background to that expected for a ∼100 mCrab source between 25 keV and 50 keV. A 6 hour observation of the Crab pulsar will differentiate between the Polar Cap/Slot Gap, Outer Gap, and Caustic models with greater than 5σ significance; and also cleanly identify the Compton reflection component in the Cygnus X-1 hard state. Long-duration flights will measure the dependence of the polarisation across the cyclotron absorption line in Hercules X-1. A scaled-down instrument will be flown as a pathfinder mission from the north of Sweden in 2010. The first science flight is planned to take place shortly thereafter.Key words: Instrumentation: detectors, Techniques: polarimetric, Pulsars: general, X-ray: binaries, Stars: neutron, Galaxies: active PACS: 95.30. Gv, 95.55.Ka, 95.55.Qf, 95.75.Hi, 95.85.Pw, 97.60Gv, 97.60.Lf, 97.80.Jp, 98.54.Cm