  This is the second in a pair of papers developing a framework to apply
logarithmic methods in the study of singular curves of genus $1$. This volume
focuses on logarithmic Gromov--Witten theory and tropical geometry. We
construct a logarithmically nonsingular moduli space of genus $1$ curves
mapping to any toric variety. The space is a birational modification of the
principal component of the Abramovich--Chen--Gross--Siebert space of
logarithmic stable maps and produces an enumerative genus $1$ curve counting
theory. We describe the non-archimedean analytic skeleton of this moduli space
and, as a consequence, obtain a full resolution to the tropical realizability
problem in genus $1$.

Algebraic Geometry