Hyperspectral imaging (HSI) enables visualisation of morphological and biochemical information, which could improve disease diagnostic accuracy. Unfortunately, the wide range of image distortions that arise during flexible endoscopy in the clinic have made integration of HSI challenging. To address this challenge, we demonstrate a hyperspectral endoscope (HySE) that simultaneously records intrinsically co-registered hyperspectral and standard-of-care white light images, which allows image distortions to be compensated computationally and an accurate hyperspectral data cube to be reconstructed as the endoscope moves in the lumen. Evaluation of HySE performance shows excellent spatial, spectral and temporal resolution and high colour fidelity. Application of HySE enables: quantification of blood oxygenation levels in tissue mimicking phantoms; differentiation of spectral profiles from normal and pathological ex vivo human tissues; and recording of hyperspectral data under freehand motion within an intact ex vivo pig oesophagus model. HySE therefore shows potential for enabling HSI in clinical endoscopy.