We report on the realisation of ultra-small-mode-volume tunable dye lasers based on hemispherical open microcavities. The cavity mode volume is of the order of cubic micrometers, such that self-diffusion of the dye molecules allows continuous wave operation over several minutes without the need for driven circulation. Such micro lasers could be integrated into lab-on-a-chip devices. A rate-equation model that incorporates the diffusion mechanism is used to predict the effect of the microcavity parameters on the lasing threshold.