The solid Earth's inner core (IC) is a sphere with a radius of about 1300 km in the center of the Earth. The information about the IC comes mainly from seismic studies. The composition of the IC is obtained by matching the seismic data and properties of candidate phases subjected to high pressure (P) and temperature (T). The close match between the density of the IC and iron suggests that the main constituent of the IC is iron. However, the stable phase of iron is still a subject of debate. One such iron phase, the body-centered cubic phase (bcc), is dynamically unstable at pressures of the IC (330-364 GPa) and low T but gets stabilized at high T characteristic of the IC (5000-7000 K). So far, ab initio molecular dynamics (AIMD) studies attempted to compute the bcc elastic properties for a small (order of 102) number of atoms. The mechanism of the bcc stabilization cannot be enabled in such cells and that has led to erroneous results. Here we apply AIMD to compute elastic moduli and sound velocities of the Fe bcc phase for a 2000 Fe atom computational cell, which is a cell of unprecedented size for ab initio calculations of iron. Unlike in previous ab initio calculations, both the longitudinal and the shear sound velocities of the Fe bcc phase closely match the properties of the IC material at P = 360 GPa and T = 6600 K, likely the PT conditions in the IC. The calculated density of the bcc iron at these PT conditions is just 3% higher than the density of the IC material according to the Preliminary Earth Model. This suggests that the widely assumed amount of light elements in the IC may need a reconsideration. The anisotropy of the bcc phase is an exact match to the most recent seismic studies.