  We report the successful synthesis of single-crystals of the layered iridate,
(Na$_{1-x}$Li$_{x}$)$_2$IrO$_3$, $0\leq x \leq 0.9$, and a thorough study of
its structural, magnetic, thermal and transport properties. The new compound
allows a controlled interpolation between Na$_2$IrO$_3$ and Li$_2$IrO$_3$,
while maintaing the novel quantum magnetism of the honeycomb Ir$^{4+}$ planes.
The measured phase diagram demonstrates a dramatic suppression of the N\'eel
temperature, $T_N$, at intermediate $x$ suggesting that the magnetic order in
Na$_2$IrO$_3$ and Li$_2$IrO$_3$ are distinct, and that at $x\approx 0.7$, the
compound is close to a magnetically disordered phase that has been sought after
in Na$_2$IrO$_3$ and Li$_2$IrO$_3$. By analyzing our magnetic data with a
simple theoretical model we also show that the trigonal splitting, on the
Ir$^{4+}$ ions changes sign from Na$_2$IrO$_3$ and Li$_2$IrO$_3$, and the
honeycomb iridates are in the strong spin-orbit coupling regime, controlled by
$\jeff=1/2$ moments.
