  We have obtained spectroscopy of Q 1343+266 AB, a pair of quasars at redshift
z = 2.03 with a projected separation of 9.5 arcseconds. This system is
well-suited for probing the Ly-alpha forest, since the two component spectra
show several Ly-alpha lines in common and several others not. Using Bayesian
statistics, under the idealization of uniform-radius spherical absorbers, we
find that the Ly-alpha cloud radius at $z \approx 1.8$ lies in the range
$40\hkpc < R < 280 \hkpc$ with $98\%$ confidence in a $(\Omega_o, \Lambda_o) =
(1, 0)$ model universe, where $h =(H_o / 100 km s^{-1}
  Mpc^{-1})$. The median value of R is 90 h^-1 kpc. \ These numbers scale up by
a factor 1.44 for $(\Omega_o, \Lambda_o) =(0.1, 0)$; and by a factor 1.85 for
(0.1, 0.9). Pressure-confined and freely-expanding Ly-alpha cloud models in
which the absorbers formed in a significantly more compressed state at $z \gg
2$ are contradicted by these new data, as are models involving stably-confined
gas concentrations in photoionization equilibrium within minihalos of cold dark
matter. The comoving density of Ly-alpha forest objects at $z\sim 2$ is $\sim
0.3 h (R/100 kpc)^{-2} Mpc^{-3}$ in a (1,0) model universe. This suggests a
possible identification of Ly-alpha clouds with the unvirialized, collapsing
progenitors of the faint, blue galaxies at $z\sim 1$.
