  We prove that the Beilinson regulator, which is a map from $K$-theory to
absolute Hodge cohomology of a smooth variety, admits a refinement to a map of
$E_\infty$-ring spectra in the sense of algebraic topology. To this end we
exhibit absolute Hodge cohomology as the cohomology of a commutative
differential graded algebra over $\mathbb{R}$. The associated spectrum to this
CDGA is the target of the refinement of the regulator and the usual $K$-theory
spectrum is the source. To prove this result we compute the space of maps from
the motivic $K$-theory spectrum to the motivic spectrum that represents
absolute Hodge cohomology using the motivic Snaith theorem. We identify those
maps which admit an $E_\infty$-refinement and prove a uniqueness result for
these refinements.
