  We consider the bond percolation model on the lattice $\mathbb{Z}^d$ ($d\ge
2$) with the constraint to be fully connected. Each edge is open with
probability $p\in(0,1)$, closed with probability $1-p$ and then the process is
conditioned to have a unique open connected component (bounded or unbounded).
The model is defined on $\mathbb{Z}^d$ by passing to the limit for a sequence
of finite volume models with general boundary conditions. Several questions and
problems are investigated: existence, uniqueness, phase transition, DLR
equations. Our main result involves the existence of a threshold $0<p^*(d)<1$
such that any infinite volume process is necessary the vacuum state in
subcritical regime (no open edges) and is non trivial in the supercritical
regime (existence of a stationary unbounded connected cluster). Bounds for
$p^*(d)$ are given and show that it is drastically smaller than the standard
bond percolation threshold in $\mathbb{Z}^d$. For instance $0.128<p^*(2)<0.202$
(rigorous bounds) whereas the 2D bond percolation threshold is equal to $1/2$.
