We develop a variational theory for geodesics joining a point and a one dimensional submanifold of a sub-Riemannian manifold. Given a Riemannian manifold ( M, g), a smooth distribution Delta subset of TM of codimension one in M, a point p is an element of M, and a smooth immersion gamma: R --> M with closed image in M and which is everywhere transversal to Delta, we look for curves in M that are stationary with respect to the Riemannian energy functional among all of the absolutely continuous curves horizontal with respect to Delta and that join p and gamma. If (M, g) is complete, such extremizers exist, and they are curves of class C-2 characterized as the solutions of an integro-differential equation or by a system of ordinary differential equations. We present some results concerning a sort of exponential map relative to the integro-differential equation and some applications. In particular, we obtain that if p and gamma are sufficiently close in M, then there exists a unique length minimizer. We obtain existence and multiplicity results by means of the Ljusternik-Schnirelman theory.