We present a subdivision scheme for the construction of 3D minimal-energy curves of given length that satisfy endpoint constraints. When given desired positions and tangents for the endpoints, and the length of the curve, the scheme iteratively builds up a minimal-energy curve. During each iteration the algorithm solves a low-dimensional optimization problem, whereby the energy of the curve is lowered and at the same time the endpoint constraints are satisfied. The energy of the curve is defined as the integral of the curvature squared and the torsion squared. With this energy function, minimal-energy curves correspond to stable configurations of flexible inextensible wires. A curve is represented by segments of piecewise constant curvature and torsion. The representation is adaptive in the sense that the number of parameters automatically varies with the complexity of the underlying curve. This scheme has been implemented and simulation results show that it typically quickly converges to very smooth curves. Our minimal-energy curve framework can be extended to minimal-energy curves of fixed length that pass through several control points and tangents. This work has applications in modeling flexible inextensible wires such as surgical sutures.