We report the successful synthesis of single-crystals of the layered iridate, (Na1−xLix)2IrO3, 0 ≤ x ≤ 0.9, and a thorough study of its structural, magnetic, thermal and transport properties. The new compound allows a controlled interpolation between Na2IrO3 and Li2IrO3, while maintaing the novel quantum magnetism of the honeycomb Ir 4+ planes. The measured phase diagram demonstrates a suppression of the Néel temperature, TN , at an intermediate x indicating that the magnetic order in Na2IrO3 and Li2IrO3 are distinct. X-ray data shows that for x ≈ 0.7 when TN is suppressed the most, the honeycomb structure is least distorted, suggesting at this intermediate doping that the material is closest to the spin liquid that has been sought after in Na2IrO3 and Li2IrO3. By analyzing our magnetic data with a single-ion theoretical model we also show that the trigonal splitting, on the Ir 4+ ions changes sign from Na2IrO3 to Li2IrO3.