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, {\it 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 {\it circular}, short period {\bf 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.
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