We propose and use a technique to measure the transverse emittance of a laser-wakefield accelerated beam of relativistic electrons. The technique is based on the simultaneous measurements of the electron beam divergence given by v(perpendicular to)/v(parallel to), the measured spectrum gamma, and the transverse electron bunch size in the bubble r(perpendicular to). The latter is obtained via the measurement of the source size of the x rays emitted by the accelerating electron bunch in the bubble. We measure a normalized rms beam transverse emittance <0.5 pi mm mrad as an upper limit for a spatially Gaussian, spectrally quasimonoenergetic electron beam with 230 MeV energy in agreement with numerical modeling and analytic theory in the bubble regime.