This paper presents stabilisation of a relative equilibrium of an underactuated autonomous underwater vehicle (AUV) with three independent control inputs on Lie group SE(3), since SE(3) can describe the evolution of a rigid body without singularities or ambiguities. First, by using the interconnection and damping assignment (IDA) method through kinetic energy shaping, a controller is proposed to stabilise the relative equilibrium in the sense of Lyapunov stability in cotangent bundle T*SE(3) modulo an isotropy subgroup when a marine vehicle immerses in an ideal fluid. Then, for an AUV submersed in a viscous fluid, an asymptotical static control law consisting of three terms is designed to stabilise the relative equilibrium via the IDA method, which these three control terms are derived from kinetic energy shaping, potential energy shaping and damping reassignment, respectively. The control term derived from kinetic energy shaping preserves stability of this motion modulo translations along the shortest semi-axis in any direction and rotations about the same axis. Moreover, the static controller is developed to a dynamic one via adaptive neural network approach, which is robust for uncertainties in the system's damping and mass. Simulation results complete our work.