  Weyl geometry is a natural extension of conformal geometry with Weyl
covariance mediated by a Weyl connection. We generalize the Fefferman-Graham
(FG) ambient construction for conformal manifolds to a corresponding
construction for Weyl manifolds. We first introduce the Weyl-ambient metric
motivated by the Weyl-Fefferman-Graham (WFG) gauge. From a top-down
perspective, we show that the Weyl-ambient space as a pseudo-Riemannian
geometry induces a codimension-2 Weyl geometry. Then, from a bottom-up
perspective, we start from promoting a conformal manifold into a Weyl manifold
by assigning a Weyl connection to the principal $\mathbb{R}_+$-bundle realizing
a Weyl structure. We show that the Weyl structure admits a well-defined initial
value problem, which determines the Weyl-ambient metric. Through the
Weyl-ambient construction, we also investigate Weyl-covariant tensors on the
Weyl manifold and define extended Weyl-obstruction tensors explicitly.
