The objective of isogeometric analysis is to address design and analysis using exactly the same representations, for both the geometry and the solution. Thus, the Lagrange polynomials usually used for interpolation in the finite element method are replaced by B-Splines functions. However, this approach is not well adapted to hyperbolic conservation laws, that require to introduce stabilization terms which are tedious to define in this context. Therefore, we present in this work a new Discontinuous Galerkin (DG) method for the numerical resolution of hyperbolic equations, that allows an exact representation of B-Spline boundaries. The method relies on the construction of local Bernstein bases from the B-Spline boundary description, Gauss-Legendre formulas to approximate the different integrals and a temporal integration using Runge-Kutta schemes. We use a Lax-Friedrichs scheme to calculate the numerical flux. Numerical results demonstrate the high accuracy of the proposed method.