  Let $p>2$ be a prime number. Let $G:=GL_2(Q_p)$ and $\pi$, $\tau$ smooth
irreducible representations of $G$ on $\bar{F}_p$-vector spaces with a central
character. We show if $\pi$ is supersingular then $Ext^1_G(\tau,\pi)\neq 0$
implies $\tau\cong \pi$. This answers affirmatively for $p>2$ a question of
Colmez. We also determine $Ext^1_G(\tau,\pi)$, when $\pi$ is the Steinberg
representation. As a consequence of our results combined with those already in
the literature one knows $Ext^1_G(\tau,\pi)$ for all irreducible
representations of $G$.
