The new compound BaGa4Se7 has been synthesized for the first time. It crystallizes in the monoclinic space group Pc with a = 7.6252 (15) angstrom, b =6.5114 (13) angstrom, c = 14.702 (4) angstrom, beta = 121.24 (2)degrees, and Z= 2. In the structure, GaSe4 tetrahedra share corners to form a three-dimensional framework with cavities occupied by Ba2+ cations. The material is a wide-band gap semiconductor with the visible and IR optical absorption edges being 0.47 and 18.0 mu m, respectively. BaGa4Se7 melts congruently at 968 degrees C and exhibits a second harmonic generation response at 1 mu m that is approximately 2-3 times that of the benchmark material AgGaS2. A first-principles calculation of the electronic structure, linear and nonlinear optical properties of BaGa4Se7 was performed. The calculated birefractive index Delta n = 0.08 at 1 mu m and the major SHG tensor elements are: d(11) =18.2 pm/V and d(13) = 20.6 pm/V. This new material is a very promising NLO crystal for practical application in the IR region.