We present results from a search for gravitational-wave bursts in the data collected by the LIGO and Virgo detectors between July 7, 2009 and October 20, 2010: data are analyzed when at least two of the three LIGO-Virgo detectors are in coincident operation, with a total observation time of 207 days. The analysis searches for transients of duration $\lesssim 1\text{ s}$ over the frequency band 64-5000 Hz, without other assumptions on the signal waveform, polarization, direction or occurrence time. All identified events are consistent with the expected accidental background. We set frequentist upper limits on the rate of gravitational-wave bursts by combining this search with the previous LIGO-Virgo search on the data collected between November 2005 and October 2007. The upper limit on the rate of strong gravitational-wave bursts at the Earth is 1.3 events per year at 90% confidence. We also present upper limits on source rate density per year and $\text{Mpc}^3$ for sample populations of standard-candle sources. As in the previous joint run, typical sensitivities of the search in terms of the root-sum-squared strain amplitude for these waveforms lie in the range $\sim 5\times 10^{-22} \text{ Hz}^{-1/2}$ to $\sim 1 \times 10^{-20} \text{ Hz}^{-1/2}$. The combination of the two joint runs entails the most sensitive all-sky search for generic gravitational-wave bursts and synthesizes the results achieved by the initial generation of interferometric detectors.