  M. Mahowald, in his work on $b{\rm o}$-resolutions, constructed a $b{\rm
o}$-module splitting of the spectrum $b{\rm o} \wedge b{\rm o}$ into a wedge of
summands related to integral Brown-Gitler spectra. In this paper, a similar
splitting of $b{\rm o} \sm tmf$ is constructed. This splitting is then used to
understand the $b{\rm o}_*$-algebra structure of $b{\rm o}_* tmf$ and allows
for a description of $b{\rm o}^* tmf$.
