  The double-perovskite compounds Sr$_2$CuTeO$_6$ and Sr$_2$CuWO$_6$ are
magnetically described as quasi-two-dimensional spin-1/2 square-lattice
$J_1{-}J_2$ Heisenberg antiferromagnets with predominant $J_1$ and $J_2$
exchange interactions, respectively. We report the low-temperature magnetic
properties of Sr$_2$CuTe$_{1-x}$W$_x$O$_6$ with randomness in the magnitudes of
$J_1$ and $J_2$. It was found that the low-temperature specific heat for
$0.1\leq x \leq 0.5$ has a large component proportional to the temperature $T$
above 1.2 K, although the low-temperature specific heat for the two parent
systems is approximately proportional to $T^3$. With decreasing temperature
below 1.2 K, the $T$-linear component decreases rapidly toward zero, which is
insensitive to the magnetic field up to 9 T. This is suggestive of the singlet
excitation decoupled from the magnetic field. The NMR spectrum for $x=0.2$
exhibits no long-range order down to 1.8 K. These results indicates that the
ground state of Sr$_2$CuTe$_{1-x}$W$_x$O$_6$ is a valence-bond-glass state with
singlet gaps.
