Recently, halide perovskite materials have shown great potential in photodetection due to their large light absorption coefficient and long carrier diffusion length. In contrast to the reported sandwich-structured devices and single-junction perovskite photodetectors, we demonstrate here a MAPbBr/MAPbCl bulk heterostructure by splicing a MAPbBr single crystal and a MAPbCl single crystal. Interestingly, the photo-carriers generated from MAPbBr and MAPbCl can only be extracted when the device is forward and reverse biased, respectively. The capability to read out the photocurrents from the single crystals with different band gaps separately enables dual-band optical detection. In particular, the heterostructure photodetector exhibits a fast on-off switching time on the order of microseconds and peak responsivities of 17 mA W at 450 nm and 7 mA W at 355 nm, respectively. Dual-band photodetection is proved to be useful for dual-wavelength light communications. Our work provides a facile route to construct bulk heterostructures and the wavelength/bias-sensitive rectifier could provide guidance for optical communication modules in the future.