We report an integral-field spectroscopic (IFS) observation of a gravitationally lensed spiral galaxy A1689B11 at redshift z = 2.54. It is the most ancient spiral galaxy discovered to date and the second kinematically confirmed spiral at z≳ 2. Thanks to gravitational lensing, this is also by far the deepest IFS observation with the highest spatial resolution (∼400 pc) on a spiral galaxy at a cosmic time when the Hubble sequence is about to emerge. After correcting for a lensing magnification of 7.2 ± 0.8, this primitive spiral disk has an intrinsic star formation rate of 22 ± 2 M _⊙ yr^-1, a stellar mass of {10}^{9.8+/- 0.3} M _⊙, and a half-light radius of {r}_1/2=2.6+/- 0.7 {kpc}, typical of a main-sequence star-forming galaxy at z∼ 2. However, the Hα kinematics show a surprisingly tranquil velocity field with an ordered rotation ({V}_{{c}}=200+/- 12 km s^-1) and uniformly small velocity dispersions ({V}_{σ ,{mean}}=23 +/- 4 km s^-1 and {V}_{σ ,{outer} - {disk}}=15+/- 2 km s^-1). The low gas velocity dispersion is similar to local spiral galaxies and is consistent with the classic density wave theory where spiral arms form in dynamically cold and thin disks. We speculate that A1689B11 belongs to a population of rare spiral galaxies at z≳ 2 that mark the formation epoch of thin disks. Future observations with the James Webb Space Telescope will greatly increase the sample of these rare galaxies and unveil the earliest onset of spiral arms.