We report on lasing in conical microcavities, which are made out of the low-loss polymer poly (methyl methacrylate) (PMMA) doped with the dye rhodamine 6G, and directly fabricated on silicon. Including a thermal reflow step during fabrication enables a significantly reduced surface roughness, resulting in low scattering losses of the whispering gallery modes (WGMs). The high cavity quality factors (above 2·106 in passive cavities) in combination with the large oscillator strength gain material enable lasing threshold energies as low as 3 nJ, achieved by free-space excitation in the quasistationary pumping regime. Lasing wavelengths are detected in the visible wavelength region around 600 nm. Finite element simulations indicate that lasing occurs in fundamental TE/TM cavity modes, as these modes have - in comparison to higher order cavity modes - the smallest mode volume and the largest overlap with the gain material. In addition, we investigate the effect of dye concentration on lasing wavelength and threshold by comparing samples with four different concentrations of rhodamine 6G. Observations are explained by modifying the standard dye laser model.