  The work of Darmon, Pozzi, and Vonk has recently shown that the RM-values of
the Dedekind-Rademacher cocycle $J_{DR}$ are Gross-Stark units up to a
controlled torsion. In the aforementioned work, it is remarked that the
measure-valued cohomology class $\mu_{DR}$ which underlies $J_{DR}$ is the
level 1 incarnation of earlier constructions by Darmon and Dasgupta. In this
paper, we make this relationship explicit by computing a concrete cocycle
representative of $\mu_{DR}$ by tracing the construction of the cohomology
class and comparing periods of weight 2 Eisenstein series. While maintaining a
global perspective in our computations, we configure the appropriate method of
smoothing cocycles which exactly yields the $p$-adic measures of Darmon and
Dasgupta when applied to $\mu_{DR}$. These methods will also explain the
optional degree zero condition imposed in Darmon and Dasgupta's work which was
remarked upon in works of Fleischer and Liu as well as Dasgupta and Kakde.
