Simulations of encounters between pairs of hard binaries, each containing a neutron star and a main-sequence star, reveal a new formation mechanism for double pulsars in dense cores of globular clusters. In many cases, the two normal stars are disrupted to form a common envelope around the pair of neutron stars, both of which will be spun up to become millisecond pulsars. We predict that a new class of pulsars, double millisecond pulsars, will be discovered in the cores of dense globular clusters. The genesis proceeds through a short-lived double-core common envelope phase, with the envelope ejected in a fast wind. It is possible that the progenitor may also undergo a double X-ray binary phase. Any circular, short-period double pulsar found in the galaxy would necessarily come from disrupted disk clusters, unlike Hulse-Taylor class pulsars or low-mass X-ray binaries which may be ejected from clusters or formed in the galaxy.