Security proofs of quantum key distribution (QKD) systems usually assume that the users have access to source of per- fect randomness. State-of-the-art QKD systems run at frequencies in the gigahertz range, requiring a sustained gigahertz rate of generation and acquisition of quantum random numbers. In this paper, we demonstrate such a high-speed random number gen- erator. The entropy source is based on an amplified spontaneous emission from an erbium-doped fibre, which is directly acquired using a standard small form-factor pluggable module. The mod- ule connects to the field programmable gate array (FPGA) of a QKD system. A real-time randomness extractor is implemented in the FPGA, and achieves a sustained rate of 1.25 Gb/s of provably random bits.