  A hamiltonian model is solved, which satisfies all requirements for a
realistic ideal quantum measurement. The system S is a spin-$\half$, whose
$z$-component is measured through coupling with an apparatus A=M+B, consisting
of a magnet $\RM$ formed by a set of $N\gg 1$ spins with quartic infinite-range
Ising interactions, and a phonon bath $\RB$ at temperature $T$. Initially A is
in a metastable paramagnetic phase. The process involves several time-scales.
Without being much affected, A first acts on S, whose state collapses in a very
brief time. The mechanism differs from the usual decoherence. Soon after its
irreversibility is achieved. Finally the field induced by S on M, which may
take two opposite values with probabilities given by Born's rule, drives A into
its up or down ferromagnetic phase. The overall final state involves the
expected correlations between the result registered in M and the state of S.
The measurement is thus accounted for by standard quantum statistical mechanics
and its specific features arise from the macroscopic size of the apparatus.
