  $\rm{Sr_2CuTe_{0.5}W_{0.5}O_6}$ is a square-lattice magnet with
super-exchange between S=1/2 $\rm{Cu^{2+}}$ spins mediated by randomly
distributed Te and W ions. Here, using sub-K temperature and 20 $\rm{\mu}$eV
energy resolution neutron scattering experiments we show that this system
transits from a gapless disorder-induced quantum spin liquid to a new quantum
state below $\rm{T_f}$ = 1.7(1) K, exhibiting a weak frozen moment of <S>/S ~
0.1 and low energy dynamic susceptibility,${\chi}''({\hbar\omega})$ linear in
energy which is surprising for such a weak freezing in this highly fluctuating
quantum regime.
