  Starting with the periodic waves earlier constructed for the gravity Whitham
equation, we parameterise the solution curves through relative wave height, and
use a limiting argument to obtain a full family of solitary waves. The
resulting branch starts from the zero solution, traverses unique points in the
wave speed-wave height space, and reaches a singular highest wave at
$\varphi(0) = \frac{\mu}{2}$. The construction is based on uniform estimates
improved from earlier work on periodic waves for the same equation, together
with limiting arguments and a Galilean transform to exclude vanishing waves and
waves levelling off at negative surface depth. In fact, the periodic waves can
be proved to converge locally uniformly to a wave with negative tails, which is
then transformed to the desired branch of solutions. The paper also contains
some proof concerning uniqueness and continuity for signed solutions (improved
touching lemma).
