We present an approach for estimating the pose of a pinhole camera from a set of 3D lines and their corresponding line segment projections on a single image. The problem is formulated as a non-linear quadratic program on the elements of the rotation matrix in a manner that establishes direct correspondence to the Perspective-n-Point (PnP) problem. By leveraging this connection to the PnP, existing methodologies are reapplied to recover the camera rotation from a constrained quadratic program. Furthermore, a novel least squares formulation is proposed to estimate the translation of the camera. Detailed comparative experiments demonstrate that the proposed approach is robust to noise and outperforms established techniques in terms of accuracy.