  We consider weak measurements (WM) of a spin observable in quantum mechanics
beyond the usual asymptotic regime. This is done by obtaining the exact
time-dependent wave functions of the measuring apparatus for general non-ideal
measurements. Ideal strong measurements and the usual WM regime are obtained as
the two extreme limiting cases of our exact treatment. We show that in the
intermediate regime non-ideal measurements lead to "semiweak" eccentric values
that differ from the usual weak values. We further show that even in the WM
regime the exact treatment leads to a meter behavior that can be markedly
distinct from the one predicted by the usual WM formalism. We give several
illustrations and discuss an application to the distinction of different
realizations of the same density matrix.
