Super-resolution (SR) technology has emerged as an effective tool for image analysis and interpretation. However, single hyperspectral (HS) image SR remains challenging, due to the high spectral dimensionality and lack of available high-resolution information of auxiliary sources. To fully exploit the spectral and spatial characteristics, in this paper, a novel single HS image SR approach is proposed based on a spatial correlation-regularized unmixing convolutional neural network (CNN). The proposed approach takes advantage of a CNN to explore the collaborative spatial and spectral information of an HS image and infer the high-resolution abundance maps, thereby reconstructing the anticipated high-resolution HS image via the linear spectral mixture model. Moreover, a dual-branch architecture network and spatial spread transform function are employed to characterize the spatial correlation between the high- and low-resolution HS images, aiming at promoting the fidelity of the super-resolved image. Experiments on three public remote sensing HS images demonstrate the feasibility and superiority in terms of spectral fidelity, compared with some state-of-the-art HS image super-resolution methods.