Methylammonium lead iodide (CH3NH3PbI3), with the organic-inorganic hybrid perovskite (OIHP) structure, has gained tremendous research interest due to its excellent photo-electron conversion ability in the application of photovoltaics. Despite its solution processed polycrystalline thin film form in solar cells, the single crystalline counterpart may offer some incredibly novel optoelectronic functionalities. In this work, a sizable (>5 mm) and high quality CH3NH3PbI3 single crystal has been synthesized by the inverse temperature crystallization method, and a white-light photodetector of the structure glass/ITO/Ga/ CH3NH3PbI3/Au was fabricated. Overbroad photo-excitation intensities ranging from 0.1 mW/cm2 to 100 mW/cm2 using a sun-light simulator, the on-off ratio is tunable in a wide-range from 65 to 2250 at zero bias voltage. The responsivity (R) and detectivity (D*) are 36.2 mA/W and 2.68×1011 Jones respectively at a weak white-light intensity such as 0.1 mW/cm2. Both the photodetective parameters decrease with the increase of the illumination intensity. Based on impedance spectra obtained at working condition and light intensity dependent Jsc measurements, the surface trap-assist recombination may play a dominating role. The corresponding lifetime (τsurf) and resistance (Rsurf_trap) exhibit fast decays at higher illumination intensities. This fundamental study may pave the way for exploring the contribution of the surface trap-assist recombination in the CH3NH3PbI3 single crystal based photodetector. We believe it is applicable for integration in micro-photonics for sensitive and weak white-light photo-detection.