  We study the random metric space called the Brownian plane, which is closely
related to the Brownian map and is conjectured to be the universal scaling
limit of many discrete random lattices such as the uniform infinite planar
triangulation. We obtain a number of explicit distributions for the Brownian
plane. In particular, we consider, for every $r>0$, the hull of radius $r$,
which is obtained by "filling in the holes" in the ball of radius $r$ centered
at the root. We introduce a quantity $Z_r$ which is interpreted as the
(generalized) length of the boundary of the hull of radius $r$. We identify the
law of the process $(Z_r)_{r>0}$ as the time-reversal of a continuous-state
branching process starting from $+\infty$ at time $-\infty$ and conditioned to
hit $0$ at time $0$, and we give an explicit description of the process of hull
volumes given the process $(Z_r)_{r>0}$. We obtain an explicit formula for the
Laplace transform of the volume of the hull of radius $r$, and we also
determine the conditional distribution of this volume given the length of the
boundary. Our proofs involve certain new formulas for super-Brownian motion and
the Brownian snake in dimension one, which are of independent interest.
