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We study the classical XY (plane rotator) model at the Kosterlitz-Thoulessphase transition. We simulate the model using the single cluster algorithm onsquare lattices of a linear size up to L=2048.We derive the finite sizebehaviour of the second moment correlation length over the lattice sizexi_{2nd}/L at the transition temperature. This new prediction and the analogousone for the helicity modulus are confronted with our Monte Carlo data. This waybeta_{KT}=1.1199 is confirmed as inverse transition temperature. Finally weaddress the puzzle of logarithmic corrections of the magnetic susceptibilitychi at the transition temperature.Comment: Monte Carlo results for xi/L in table 1 and 2 corrected. Due to a  programming error,these numbers were wrong by about a factor 1+1/L^2.  Correspondingly the fits with L_min=64 and 128 given in table 5 and 6 are  changed by little.The central results of the paper are not affected. Wrong  sign in eq.(52) corrected. Appendix extende

The two-dimensionalXYmodel at the transition temperature: a high-precision Monte Carlo study