Old Globular Clusters (GCs) in the Milky Way have ages of about 13 Gyr, placing their formation time in the reionization epoch. We propose a novel scenario for the formation of these systems based on the merger of two or more atomic cooling halos at high-redshift (z>6). First generation stars are formed as an intense burst in the center of a minihalo that grows above the threshold for hydrogen cooling (halo mass M_h~10^8 Msun) by undergoing a major merger within its cooling timescale (~150 Myr). Subsequent minor mergers and sustained gas infall bring new supply of pristine gas at the halo center, creating conditions that can trigger new episodes of star formation. The dark-matter halo around the GC is then stripped during assembly of the host galaxy halo. Minihalo merging is efficient only in a short redshift window, set by the LCDM parameters, allowing us to make a strong prediction on the age distribution for old GCs. From cosmological simulations we derive an average merging redshift =9 and narrow distribution Dz=2, implying average GC age =13.0+/-0.2 Gyr including ~0.2 Gyr of star formation delay. Qualitatively, our scenario reproduces other general old GC properties (characteristic masses and number of objects, metallicity versus galactocentric radius anticorrelation, radial distribution), but unlike age, these generally depend on details of baryonic physics. In addition to improved age measurements, direct validation of the model at z~10 may be within reach of ultradeep gravitationally lensed observations with the James Webb Space Telescope. ; Comment: ApJL accepted, minor content changes to highlight the robust predictions for GC ages; for an animated version of Fig. 1 (minihalo mergers movie) see http://www.ph.unimelb.edu.au/~mtrenti/trenti_padoan_jimenez_2015_fig1_GC_movie.gif