  Let $G$ be a simple simply connected group scheme defined over ${\mathbb
F}_{p}$ and $k$ be an algebraically closed field of characteristic $p>0$.
Moreover, let $B$ be a Borel subgroup of $G$ and $U$ be the unipotent radical
of $B$. In this paper the authors compute the third cohomology group for $B$
and its Frobenius kernels, $B_{r}$, with coefficients in a one-dimensional
representation. These computations hold with relatively mild restrictions on
the characteristic of the field. As a consequence of our calculations, the
third ordinary Lie algebra cohomology group for ${\mathfrak u}=\text{Lie }U$
with coefficients in $k$ is determined, as well as the third $G_{r}$-cohomology
with coefficients in the induced modules $H^{0}(\lambda)$.
