Gravitational waves from a variety of sources are predicted to superpose to create a stochastic background. This background is expected to contain unique information from throughout the history of the universe that is unavailable through standard electromagnetic observations, making its study of fundamental importance to understanding the evolution of the universe. We carry out a search for the stochastic background with the latest data from LIGO and Virgo. Consistent with predictions from most stochastic gravitational-wave background models, the data display no evidence of a stochastic gravitational-wave signal. Assuming a gravitational-wave spectrum of ${\Omega}_{\rm GW}(f)={\Omega}_{\alpha}\left({f/f_{\rm{ref}}}\right)^{\alpha}$, we place 95\% confidence level upper limits on the energy density of the background in each of four frequency bands spanning 41.5--1726~Hz. In the frequency band of 41.5--169.25~Hz for a spectral index of $\alpha=0$, we constrain the energy density of the stochastic background to be $\Omega_{\rm GW}(f)<5.6\times10^{-6}$. For the 600--1000~Hz band, $\Omega_{\rm GW}(f)<0.14(f/900~{\rm Hz})^3$, a factor of 2.5 lower than the best previously reported upper limits. We find $\Omega_{\rm GW}(f)<1.8\times10^{-4}$ using a spectral index of zero for 170--600~Hz and $\Omega_{\rm GW}(f)<1.0(f/1300~{\rm Hz})^3$ for 1000--1726~Hz, bands in which no previous direct limits have been placed. The limits in these four bands are the lowest direct measurements to date on the stochastic background. We discuss the implications of these results in light of the recent claim by the BICEP2 experiment of the possible evidence for inflationary gravitational waves.