  We predict negative temperature states in the Discrete Nonlinear Sch\"odinger
equation as exact solutions of the associated Wave Kinetic equation. Those
solutions are consistent with the classical thermodynamics formalism. Explicit
calculation of the entropy as a function of the energy and number of particles
is performed analytically. Direct numerical simulations of the DNLS equation
are in agreement with theoretical results. We show that the key ingredient for
observing negative temperatures in lattices is the boundedness of the
dispersion relation in its domain. States with negative temperatures are
characterized by an accumulation of particles and energy at wavenumber $k=\pi$.
