  The lower critical field $H_{c1}$ has been carefully measured on a well
shaped cylindrical sample of the new superconductor $MgB_2$ fabricated by high
pressure synthesis. The penetration depth $\lambda$ is calculated from the
$H_{c1}$ data. It is found that a linear relation of $H_{c1}(T)$ appears in
whole temperature region below $T_c$. Furthermore a finite slope of
$dH_{c1}/dT$ and $d\lambda(T)/dT$ remains down to the lowest temperature (2 K).
These are inconsistent with the expectation for a widely thought s-wave
superconductivity in $MgB_2$.
