  We have studied the magnetic properties and spin-dynamics of the
structurally-ordered double perovskite Sr$_2$CoOsO$_6$. Our neutron
diffraction, muon spin relaxation and ac-susceptibility measurements reveal two
antiferromagnetic (AFM) phases on cooling from room temperature down to 2 K. In
the first AFM phase, with transition temperature $T_{\mathrm{N}}=108$ K, cobalt
(3$d^7$, $S=3/2$) and osmium (5$d^2$, $S=1$) moments fluctuate dynamically,
while their \textit{average} effective moments undergo long-range order. In the
second AFM phase with $T_{\mathrm{N}}=67$ K, cobalt moments first become frozen
and induce a noncollinear spin-canted AFM state, while dynamically fluctuating
osmium moments are later frozen into a randomly canted state at $T\approx 5$ K.
Our \textit{ab initio} calculations indicate that the effective exchange
coupling between cobalt and osmium sites is rather weak, so that cobalt and
osmium sublattices exhibit different ground states and spin-dynamics, making
Sr$_2$CoOsO$_6$ distinct from previously reported double-perovskite compounds.
