We propose a new finite element method for solving second order elliptic interface problems whose solution has a Robin type jump along the interface. We cast the problem into a new variational form and introduce a finite element method to solve it using a uniform grid. We modify the P-1-Crouzeix-Raviart element so that the shape functions satisfy the jump conditions along the interface. We note that there are cases that the Lagrange type basis can not be used because of the jump in the value. Numerical experiments are provided.