Digital holographic (DH) microscopy has grown into a powerful nondestructive technique for the real-time study of living cells including dynamic membrane changes and cell fluctuations in nanometer and sub-nanometer scales. The conventional DH microscopy configurations require a separately generated coherent reference wave that results in a low phase stability and a necessity to precisely adjust the intensity ratio between two overlapping beams. In this work, we present a compact, simple, and very stable common-path DH microscope, employing a self-referencing configuration. The microscope is implemented by a diode laser as the source and a Fresnel biprism for splitting and recombining the beams simultaneously. In the overlapping area, linear interference fringes with high contrast are produced. The frequency of the interference pattern could be easily adjusted by displacement of the biprism along the optical axis without a decrease in fringe contrast. To evaluate the validity of the method, the spatial noise and temporal stability of the setup are compared with the common off-axis DH microscope based on a Mach-Zehnder interferometer. It is shown that the proposed technique has low mechanical noise as well as superb temporal stability with sub-nanometer precision without any external vibration isolation. The higher temporal stability improves the capabilities of the microscope for studying micro-object fluctuations, particularly in the case of biological specimens. Experimental results are presented using red blood cells and silica microspheres to demonstrate the system performance.