  This paper is the second in a series of planned papers which provide first
bijective proofs of alternating sign matrix results. Based on the main result
from the first paper, we construct a bijective proof of the enumeration formula
for alternating sign matrices and of the fact that alternating sign matrices
are equinumerous with descending plane partitions. We are also able to refine
these bijections by including the position of the unique $1$ in the top row of
the matrix. Our constructions rely on signed sets and related notions. The
starting point for these constructions were known ``computational'' proofs, but
the combinatorial point of view led to several drastic modifications. We also
provide computer code where all of our constructions have been implemented.
