  We have studied the vortex-matter phase diagram of low C doped MgB$_2$
superconductor. The significant finding is the appearance of a peak effect,
which, as both our global magnetization and local ac susceptibility
measurements revealed, is placed well below the $H_{c2}$-line. The absence of
significant bulk pinning below the onset line $H_{on}(T)$ of the peak effect
implies that the Bragg glass phase is present for $H < H_{on}(T)$.
Interestingly, the unexpected absence of bulk pinning above the end point line
$H_{ep}(T)$ of the peak effect implies the presence of a slightly pinned vortex
phase in the regime $H_{ep}(T) < H < H_{c2}(T)$. In addition, the observed
increase of the $H_{c2}^c(0)$ since the carbon doped MgB$_2$ becomes more dirty
and the reduced anisotropy, in comparison to the pure MgB$_2$, makes the
C-doped MgB$_2$ more favorable for practical applications.
