Using quadrupole scan measurements we show laser-wakefield accelerated electrons to have a normalized transverse emittance of 0.21+0.01−0.02π mm mrad at 245 MeV. We demonstrate a multishot and a single-shot method, the mean emittance values for both methods agree well. A simple model of the beam dynamics in the plasma density downramp at the accelerator exit matches the source size and divergence values inferred from the measurement. In the energy range of 245 to 300 MeV the normalized emittance remains constant.