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Models of the absorbing ``clouds'' responsible for the \Lya\ forest observed in spectra of high-redshift quasars require some knowledge of the size of the clouds, which to-date remains poorly constrained. We present observations of the \Lya\ forest of the close quasar pair 1343+2640A ($z_{em} = 2.029$) and B ($z_{em} = 2.031$). We detect eight absorption lines of \Lya\ common to both spectra and not attributable to metal line systems in the redshift range $1.7 < z < 2.1$, and four lines which are seen in one spectrum but not the other. At the $9.5^{\prime\prime}$ separation of the two quasars, this implies a firm lower limit on the characteristic size of the \Lya\ clouds of $40\, h^{-1}_{100}$~kpc (where $h_{100} \equiv H_0/100$~km\,s$^{-1}$\,Mpc$^{-1}$, $q_0 = 0.5$) at a redshift $z\simeq 1.8$. The upper limit on the cloud size is much more uncertain due to the small number of observed lines, but taking the observed fraction of common lines at face value in the context of a simple model, the absorbers are shown have diameters smaller than about $620\, h^{-1}_{100}$~kpc. Significant velocity and equivalent width variations are seen with an rms velocity difference of $\sim65$\,\kms\ between the common absorption lines along the two lines of sight. The large extent and quiescent state of the clouds pose a serious challenge to current theoretical models of the \Lya\ absorbers.
