We define the fundamental group underlying to Lichtenbaum's Weil-\'etale
cohomology for number rings. To this aim, we define the Weil-\'etale topos as a
refinement of the Weil-\'etale sites introduced in \cite{Lichtenbaum}. We show
that the (small) Weil-\'etale topos of a smooth projective curve defined in
this paper is equivalent to the natural definition given in
\cite{Lichtenbaum-finite-field}. Then we compute the Weil-\'etale fundamental
group of an open subscheme of the spectrum of a number ring. Our fundamental
group is a projective system of locally compact topological groups, which
represents first degree cohomology with coefficients in locally compact abelian
groups. We apply this result to compute the Weil-\'etale cohomology in low
degrees and to prove that the Weil-\'etale topos of a number ring satisfies the
expected properties of the conjectural Lichtenbaum topos.