  In this article, we study a free boundary isometric embedding problem for
abstract Riemannian two-manifolds with the topology of the disc. Under the
assumption of positive Gauss curvature and geodesic curvature of the boundary
being equal to one, we show that any such disc may be isometrically embedded
into the Euclidean three space $\mathbb{R}^3$ such that the image of the
boundary meets the unit sphere $\mathbb{S}^2$ orthogonally. Moreover, we also
show that the embedding is unique up to rotations and reflections through
planes containing the origin. Finally, we define a new Brown-York type
quasi-local mass for certain free boundary surfaces and discuss its positivity.
