We report a simultaneous measurement of a longitudinal spin Seebeck effect (LSSE) and thermal conductivity in a Pt/Y3Fe5O12 (YIG)/Pt system in a temperature range from 10 to 300 K. By directly monitoring the temperature difference in the system, we excluded thermal artifacts in the LSSE measurements. It is found that both the LSSE signal and the thermal conductivity of YIG exhibit sharp peaks at the same temperature, different from previous reports. The maximum LSSE coefficient is found to be S-LSSE > 10 mu V/K, one-order-of magnitude greater than the previously reported values. The concomitant enhancement of the LSSE and thermal conductivity of YIG suggests the strong correlation between magnon and phonon transport in the LSSE.