Earth continuously generates a dipole magnetic field in its convecting liquid outer core by a self-sustained dynamo action. Metallic iron is a dominant component of the outer core, so its electrical and thermal conductivity controls the dynamics and thermal evolution of Earth's core1. However, in spite of extensive research, the transport properties of iron under core conditions are still controversial2,3,4,5,6,7,8,9. Since free electrons are a primary carrier of both electric current and heat, the electron scattering mechanism in iron under high pressure and temperature holds the key to understanding the transport properties of planetary cores. Here we measure the electrical resistivity (the reciprocal of electrical conductivity) of iron at the high temperatures (up to 4,500 kelvin) and pressures (megabars) of Earth's core in a laser-heated diamond-anvil cell. The value measured for the resistivity of iron is even lower than the value extrapolated from high-pressure, low-temperature data using the Bloch-Grüneisen law, which considers only the electron-phonon scattering. This shows that the iron resistivity is strongly suppressed by the resistivity saturation effect at high temperatures. The low electrical resistivity of iron indicates the high thermal conductivity of Earth's core, suggesting rapid core cooling and a young inner core less than 0.7 billion years old10. Therefore, an abrupt increase in palaeomagnetic field intensity around 1.3 billion years ago11may not be related to the birth of the inner core.