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We report a colloidal synthesis of blue emissive, stable cube-shaped CsPbBr3 quantum dots (QDs) in the strong quantum confinement regime via a dissolution-recrystallization starting from pre-synthesized (KxCs1-x)4PbBr6 nanocrystals which are then reacted with PbBr2. This is markedly different from the known case of Cs4PbBr6 nanocrystals that react within seconds with PbBr2 and get transformed into much larger, green emitting CsPbBr3 nanocrystals. Here, instead, the conversion of (KxCs1-x)4PbBr6 nanocrystals to CsPbBr3 QDs occurs in a time span of hours, and tuning of the QDs size is achieved by adjusting the concentration of precursors. The QDs exhibit excitonic features in optical absorption that are tunable in the 420 - 452 nm range, accompanied by blue photoluminescence with quantum yield around 60%. Detailed spectroscopic investigations in both the single and multi-exciton regime reveal the exciton fine structure and the effect of Auger recombination of these CsPbBr3 QDs, confirming theoretical predictions for this system.