  We study the limiting behavior of the discrete spectra associated to the
principal congruence subgroups of a reductive group over a number field. While
this problem is well understood in the cocompact case (i.e., when the group is
anisotropic modulo the center), we treat groups of unbounded rank. For the
groups GL(n) and SL(n) we show that the suitably normalized spectra converge to
the Plancherel measure (the limit multiplicity property). For general reductive
groups we obtain a substantial reduction of the problem. Our main tool is the
recent refinement of the spectral side of Arthur's trace formula obtained in
[FLM11, FL11], which allows us to show that for GL(n) and SL(n) the
contribution of the continuous spectrum is negligible in the limit.
