  We show that the partition function of the multi-layer semi-discrete directed
polymer converges in the intermediate disorder regime to the partition function
for the multi-layer continuum polymer introduced by O'Connell and Warren. This
verifies, modulo a previously hidden constant, an outstanding conjecture
proposed by Corwin and Hammond. A consequence is the identification of the KPZ
line ensemble as logarithms of ratios of consecutive layers of the continuum
partition function. Other properties of the continuum partition function, such
as continuity, strict positivity and contour integral formulas to compute mixed
moments, are also identified from this convergence result.
