  The single crystalline form of an insulator, Sr$_3$CuIrO$_{6+\delta}$, is
shown to exhibit unexpectedly more than one magnetic transition (at 5 and 19 K)
with spin-glass-like magnetic susceptibility behaviour. On the basis of this
finding, viz., inhomogeneous magnetism in a chemically homogeneous material, we
propose that the idea of "phase- separation" described for manganites [1] is
more widespread in different ways. The observed experimental features enable us
to make a comparison with the predictions of a recent toy model [2] on {\it
magnetic} phase separation in an insulating environment.
