Proof of the existence of a significant population of normal disk galaxies at redshift z > 2 would have profound implications for theories of structure formation and evolution .
We present evidence based on Keck HIRES spectra that the damped Ly \alpha absorber at z = 3.15 toward the quasar Q 2233+1310 may well be such an example .
Djorgovski et al .
have recently detected the Ly \alpha emission from the absorber , which we assume is at the systemic redshift of the absorbing galaxy .
By examining the profiles of the metal absorption lines arising from the absorbing galaxy in relation to its systemic redshift , we find strong kinematical evidence for rotation .
Therefore the absorber is likely to be a disk galaxy .
The inferred circular velocity for the galaxy is \geq 200 km s ^ { -1 } .
With a separation of \simeq 17 kpc ( q _ { 0 } = 0.1 , H _ { 0 } = 75 ) between the galaxy and the quasar sightline , the implied dynamic mass for the galaxy is \geq 1.6 \times 10 ^ { 11 } M _ { \odot } .
The metallicity of the galaxy is found to be [ Fe/H ] = -1.4 , typical of damped Ly \alpha galaxies at such redshifts .
However , in another damped Ly \alpha absorber at z = 2.81 toward Q 0528 - 2505 , no kinematical evidence for galactic rotation is evident .
In the latter case , the damped Ly \alpha absorber occurs near the background quasar in redshift so its properties may be influenced by the background quasar .
These represent the only two cases at present for which the technique used here may be applied .
Future applications of the same technique to a large sample of damped Ly \alpha galaxies may allow us to determine if a significant population of disk galaxies already existed only a few billion years after the Big Bang .