Short Gamma-Ray Bursts (SGRBs) are among the most luminous explosions in the universe, releasing in less than one second the energy emitted by our Galaxy over one year. Despite decades of observations, the nature of their ``central-engine'' remains unknown. I will show how the dynamics of a binary of magnetized neutron stars leads to a rapidly-spinning black hole surrounded by a hot and highly-magnetized torus. The development of magnetohydrodynamical instabilities in the torus can amplify by several orders of magnitude the initially turbulent magnetic field, yielding an ordered poloidal field of ~ 1015 G along the black-hole spin-axis, within a half-opening angle of 30 deg, which may naturally launch a relativistic jet. I will show that the broad consistency of these ab-initio calculations with SGRB observations provides important confirmations that the merger of magnetized neutron stars can provide the basic physical conditions for the central-engine of SGRBs.