  An essentially-exact approach to compute the wavefunction in the
time-dependent many-boson Schr\"odinger equation is derived and employed to
study accurately the process of splitting a trapped condensate when ramping-up
a barrier such that a double-well is formed. We follow the role played by
many-body excited states during the splitting process. Among others, a
'counter-intuitive' regime is found in which the evolution of the condensate
when the barrier is ramped-up sufficiently slow {\it is not} to the
ground-state which is a fragmented condensate, but to a low-lying excited-state
which is a coherent condensate. Experimental implications are discussed.
