An etch method based on surface tension driven flows of hydrofluoric acid microdroplets for the fabrication of low-loss, subwavelength-diameter biconical fiber tapers is presented. Tapers with losses less than 0.1 dB/mm are demonstrated, corresponding to an order of magnitude increase in the optical transmission over previous acid-etch techniques. The etch method produces adiabatic taper transitions with minimal surface corrugations. A biconical fiber taper fabricated using this method is used to demonstrate an erbium doped silica microsphere laser.