Let D be an integral domain, S be a torsion-free grading monoid such that the quotient group of S is of type (0, 0, 0, ${ldots}$), and D[S] be the semigroup ring of S over D. We show that D[S] is an H-domain if and only if D is an H-domain and each maximal t-ideal of S is a $v$-ideal. We also show that if $mathbb{R}$ is the eld of real numbers and if ${Gamma}$ is the additive group of rational numbers, then $mathbb{R}[{Gamma}]$ is not an H-domain.