VERITAS has been monitoring the very-high-energy (VHE; > 100 GeV) gamma-ray activity of the radio galaxy M 87 since 2007. During 2008, flaring activity on a timescale of a few days was observed with a peak flux of (0.70 ± 0.16) × 10^–11 cm^–2 s^–1 at energies above 350 GeV. In 2010 April, VERITAS detected a flare from M 87 with peak flux of (2.71 ± 0.68) × 10^–11 cm^–2 s^–1 forE > 350 GeV. The source was observed for six consecutive nights during the flare, resulting in a total of 21 hr of good-quality data. The most rapid flux variation occurred on the trailing edge of the flare with an exponential flux decay time of 0.90^+0.22 _–0.15 days. The shortest detected exponential rise time is three times as long, at 2.87^+1.65 _–0.99 days. The quality of the data sample is such that spectral analysis can be performed for three periods: rising flux, peak flux, and falling flux. The spectra obtained are consistent with power-law forms. The spectral index at the peak of the flare is equal to 2.19 ± 0.07. There is some indication that the spectrum is softer in the falling phase of the flare than the peak phase, with a confidence level corresponding to 3.6 standard deviations. We discuss the implications of these results for the acceleration and cooling rates of VHE electrons in M 87 and the constraints they provide on the physical size of the emitting region.