It was recently demonstrated that in La2-xSrxCuO4 the magnetic-field (H) dependence of the low-temperature thermal conductivity kappa up to 16 T reflects whether the normal state under high magnetic field is a metal or an insulator. We measure the H dependence of kappa in YBa(2)Cu(3)O(y) (YBCO) at subkelvin temperatures for a wide doping range, and find that at low doping the kappa(H) behavior signifies the change in the ground state in this system as well. Surprisingly, the critical doping is found to be located deeply inside the underdoped region, about the hole doping of 0.07 hole/Cu; this critical doping is apparently related to the stripe correlations as revealed by the in-plane resistivity anisotropy.