  We show that a proper degeneracy at $q=0$ of the $q$-deformed rook monoid of
Solomon is the algebra of a monoid $R_n^0$ namely the $0$-rook monoid, in the
same vein as Norton's $0$-Hecke algebra being the algebra of a monoid $H_n^0 =
H^0(A_{n-1})$ (in Cartan type~$A_{n-1}$). As expected, $R_n^0$ is closely
related to the latter: it contains the $H^0(A_{n-1})$ monoid and is a quotient
of $H^0(B_{n})$. We give a presentation for this monoid as well as a
combinatorial realization as functions acting on the classical rook monoid
itself. On the way we get a Matsumoto theorem for the rook monoid a result
which was conjectured by Solomon.
  The $0$-rook monoid shares many combinatorial properties with the Hecke
monoid: its Green right preorder is an actual order, and moreover a lattice
(analogous to the right weak order) which has some nice combinatorial, and
geometrical features. In particular the $0$-rook monoid is J-trivial.
  Following Denton-Hivert-Schilling-Thi\'ery, it allows us to describe its
representation theory including the description of the simple and projective
modules. We further show that $R_n^0$ is projective on $H_n^0$ and make
explicit the restriction and induction functors along the inclusion map. We
finally give a (partial) associative tower structures on the family of
$(R_n^0)$ and we discuss its representation theory.
