  We report the results of ac and dc magnetization (M) and heat-capacity (C)
measurements on the solid solution, Sr$_3$Cu$_{1-x}$Zn$_x$IrO$_6$. While the Zn
end member is known to form in a rhombohedral pseudo one-dimensional
K$_4$CdCl$_6$ structure with an antiferromagnetic ordering temperature of
(T$_N$ =) 19 K, the Cu end member has been reported to form in a monoclinically
distorted form with a Curie temperature of (T$_C$ =) 19 K. The magnetism of the
Zn compound is found to be robust to synthetic conditions and is broadly
consistent with the behavior known in the literature. However, we find a lower
magnetic ordering temperature (T$_o$) for our Cu compound (~ 13 K), thereby
suggesting that T$_o$ is sensitive to synthetic conditions. The Cu sample
appears to be in a spin-glass-like state at low temperatures, judged by a
frequency dependence of ac magnetic susceptibility and a broadening of the C
anomaly at the onset of magnetic ordering, in sharp contrast to earlier
proposals. Small applications of magnetic field, however, drive this system to
ferromagnetism as inferred from the M data. Small substitutions for Cu/Zn (x =
0.75 or 0.25) significantly depress magnetic ordering; in other words, T$_o$
varies non-monotonically with x (T$_o$ ~ 6, 3 and 4 K for x = 0.25, 0.5, and
0.67 respectively). The plot of inverse susceptibility versus temperature is
non-linear in the paramagnetic state as if correlations within (or among) the
magnetic chains continuously vary with temperature. The results establish
