Joining niobium and stainless steel is important for lowering the cost of Superconducting Radio-Frequency (SRF) cavities and thus attracting great interests in the superconducting accelerator community. In this article, we reported a vacuum brazing technique between reactor grade niobium and 316L stainless steel using oxygen-free copper as filler material. Microstructure and mechanical properties at different temperature of both samples and transition joints were investigated. The microstructure demonstrated that brittle intermetallic layer was formed and the diffusion phenomenon was observed. In spite of the existence of brittle intermetallic, the brazing joints’ application in SRF cavities was not affected. Mechanical tests, including tensile tests, shear tests were conducted both at room temperature (300 K) and liquid nitrogen temperature (77 K). At liquid nitrogen temperature, the tensile strengths are higher than those at the room temperature, while the shear strengths are reduced due to the formation of brittle intermetallic. The transition joints always have higher mechanical strengths than samples when the thickness of the welding seam was strictly controlled. Leak tests were also performed on the transition. The joints’ leak rates are all lower than 1.1×10_^-11 mbar.L/s, indicating the vacuum brazing technology is applicable to high vacuum vessels used in cryogenic temperature, such as stainless steel helium vessel for SRF cavities.