Heat management has become more and more critical, especially in miniaturized modern devices, so the exploration of highly thermally conductive materials with electrical insulation is of great importance. Here, we report that high-quality one-atom-thin hexagonal boron nitride (BN) has a thermal conductivity (κ) of 751 W/mK at room temperature, the second largest κ per unit weight among all semiconductors and insulators. The κ of atomically thin BN decreases with increased thickness. Our molecular dynamic simulations accurately reproduce this trend, and the density functional theory (DFT) calculations reveal the main scattering mechanism. The thermal expansion coefficients of monolayer to trilayer BN at 300 to 400 K are also experimentally measured for the first time. Owing to its wide bandgap, high thermal conductivity, outstanding strength, good flexibility, and excellent thermal and chemical stability, atomically thin BN is a strong candidate for heat dissipation applications, especially in the next generation of flexible electronic devices.