  Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely
classified in a series of three papers. This second one applies the powerful
tool of trigonometric Diophantine equations to classify the case of
$a^3b$-quadrilaterals with all angles being rational degrees. There are $12$
sporadic and $3$ infinite sequences of quadrilaterals admitting the $2$-layer
earth map tilings together with their modifications, and $3$ sporadic
quadrilaterals admitting $4$ exceptional tilings. Among them only $3$
quadrilaterals are convex. New interesting non-edge-to-edge triangular tilings
are obtained as a byproduct.
