  Let $A={\mathbb F}_q[t]$ be the polynomial ring over a finite field ${\mathbb
F}_q$ and let $\phi $ and $\psi$ be $A-$Drinfeld modules. In this paper we
consider the group ${\mathrm{Ext}}^1(\phi ,\psi )$ with the Baer addition. We
show that if $\mathrm{rank}\phi >\mathrm{rank}\psi$ then
$\mathrm{Ext^1}(\phi,\psi)$ has the structure of a \tm module. We give complete
algorithm describing this structure. We generalize this to the cases:
$\mathrm{Ext^1}(\Phi,\psi)$ where $\Phi$ is a \tm module and $\psi$ is a
Drinfeld module and $\mathrm{Ext^1}(\Phi, C^{\otimes e})$ where $\Phi$ is a \tm
module and $C^{\otimes e}$ is the $e$-th tensor product of Carlitz module. We
also establish duality between $\Ext$ groups for \tm modules and the
corresponding adjoint ${\mathbf t}^{\sigma}$-modules. Finally, we prove the
existence of $"\Hom-\Ext"$ six-term exact sequences for \tm modules and dual
\tm motives. As the category of \tm modules is only additive (not abelian) this
result is nontrivial.
