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The $\beta$-Ga$_2$O$_3$ has exceptional electronic properties with vast potential in power and RF electronics. Despite the excellent demonstrations of high-performance unipolar devices, the lack of p-type doping in $\beta$-Ga$_2$O$_3$ has hindered the development of Ga$_2$O$_3$-based bipolar devices. The approach of p-n diodes formed by polycrystalline p-type oxides with n-type $\beta$-Ga$_2$O$_3$ can face severe challenges in further advancing the $\beta$-Ga$_2$O$_3$ bipolar devices due to their unfavorable band alignment and the poor p-type oxide crystal quality. In this work, we applied the semiconductor grafting approach to fabricate monocrystalline Si/$\beta$-Ga$_2$O$_3$ p-n diodes for the first time. With enhanced concentration of oxygen atoms at the interface of Si/$\beta$-Ga$_2$O$_3$, double side surface passivation was achieved for both Si and $\beta$-Ga$_2$O$_3$ with an interface Dit value of 1-3 x 1012 /cm2 eV. A Si/$\beta$-Ga$_2$O$_3$ p-n diode array with high fabrication yield was demonstrated along with a diode rectification of 1.3 x 107 at +/- 2 V, a diode ideality factor of 1.13 and avalanche reverse breakdown characteristics. The diodes C-V shows frequency dispersion-free characteristics from 10 kHz to 2 MHz. Our work has set the foundation toward future development of $\beta$-Ga$_2$O$_3$-based transistors.