  We address the puzzling observational indications for very "cold" galactic
discs at redshifts $z \gtrsim 3$, an epoch when discs are expected to be highly
perturbed. Using a high-resolution cosmological zoom-in simulation, we identify
such a cold disc at $z\sim 3.5$, with a rotation velocity to velocity
dispersion ratio of $v_\phi/\sigma_r \simeq 5$ for the total gas. It forms as a
result of a period of intense accretion of co-planar, co-rotating gas via cold
cosmic-web streams. This thin disc survives for $\sim 5$ orbital periods, after
which it is disrupted by mergers and counter-rotating streams, longer but
consistent with our estimate that a galaxy of this mass
($M_\star\sim10^{10}\mathrm{M_\odot}$) typically survives merger-driven spin
flips for $\sim 2-3$ orbital periods. We find that $v_\phi/\sigma_r$ is highly
sensitive to the tracer used to perform the kinematic analysis. While it is
$v_\phi/\sigma_r \simeq 3.5$ for atomic HI gas, it is $v_\phi/\sigma_r \simeq
8$ for molecular CO and H$_2$. This reflects the confinement of molecular gas
to cold, dense clouds that reside near the disc mid-plane, while the atomic gas
is spread into a turbulent and more extended thicker disc.
