  An exactly solvable model for a quantum measurement is discussed which is
governed by hamiltonian quantum dynamics. The $z$-component $\hat s_z$ of a
spin-1/2 is measured with an apparatus, which itself consists of magnet coupled
to a bath. The initial state of the magnet is a metastable paramagnet, while
the bath starts in a thermal, gibbsian state. Conditions are such that the act
of measurement drives the magnet in the up or down ferromagnetic state
according to the sign of $s_z$ of the tested spin. The quantum measurement goes
in two steps. On a timescale $1/\sqrt{N}$ the off-diagonal elements of the
spin's density matrix vanish due to a unitary evolution of the tested spin and
the $N$ apparatus spins; on a larger but still short timescale this is made
definite by the bath. Then the system is in a `classical' state, having a
diagonal density matrix. The registration of that state is a quantum process
which can already be understood from classical statistical mechanics. The von
Neumann collapse and the Born rule are derived rather than postulated.
