  We extend the Framization of the Temperley-Lieb algebra to Coxeter systems of
type $\mathtt{B}$. We first define a natural extension of the classical
Temperley-Lieb algebra to Coxeter systems of type $\mathtt{B}$ and prove that
such an extension supports a unique linear Markov trace function. We then
introduce the Framization of the Temperley-Lieb algebra of type $\mathtt{B}$ as
a quotient of the Yokonuma-Hecke algebra of type $\mathtt{B}$. The main theorem
provides necessary and sufficient conditions for the Markov trace defined on
the Yokonuma-Hecke algebra of type $\mathtt{B}$ to pass to the quotient
algebra. Using the main theorem, we construct invariants for framed links and
classical links inside the solid torus.
