WEAVE is the next-generation optical spectroscopy facility for the William Herschel Telescope and aims at spectroscopic follow-up of ground-based (LOFAR) and space-based (Gaia) surveys. WEAVE places in the re-fitted prime focus either 1000 fibres, 20 fibre-coupled mini-IFUs or a single large 600 fibre IFU. A spectrograph on the Nasmyth platform analyses the light and supports, in a single exposure, either R~5,000 observations over the full 366- 975 nm wavelength range or simultaneous R~20,000 observations over two out of three pre-specified bands within this wavelength range. This paper describes the requirements, optical design and mechanical design of the WEAVE spectrograph.