  In this paper, we study the Dirichlet problem for the minimal surface
equation in $\rm Sol_3$ with possible infinite boundary data, where $\rm Sol_3$
is the non-abelian solvable $3$-dimensional Lie group equipped with its usual
left-invariant metric that makes it into a model space for one of the eight
Thurston geometries. Our main result is a Jenkins-Serrin type theorem which
establishes necessary and sufficient conditions for the existence and
uniqueness of certain minimal Killing graphs with a non-unitary Killing vector
field in $\rm Sol_3$.
