An exact Lagrangian submanifold \(L\) in the symplectization of standard contact \((2n-1)\)-space with Legendrian boundary \(\Sigma\) can be glued to itself along \(\Sigma\). This gives a Legendrian embedding \(\Lambda(L,L)\) of the double of \(L\) into contact \((2n+1)\)-space. We show that the Legendrian isotopy class of \(\Lambda(L,L)\) is determined by formal data: the manifold \(L\) together with a trivialization of its complexified tangent bundle. In particular, if \(L\) is a disk then \(\Lambda(L,L)\) is the Legendrian unknot.