The temperature dependences of magnetic susceptibility, the specific heat and the thermal expansion of β-Mn1-x Osx alloys have been investigated. The electronic specific heat coefficient γ is significantly large and drastically decreases with increasing x. The specific heat peak associated with the Néel temperature TN is clearly observed in high Os concentration regions. On the other hand, in low Os concentration regions, the peak is not clear and a linear relation between γ and TN3/4 is observed. Therefore, it is concluded that β-Mn1-xOsx alloys with low x are weak itinerant-electron antiferromagnets. The thermal expansion coefficient α in the paramagnetic state is remarkably large in the low Os concentration regions and becomes smaller with increasing x; that is, the larger the electronic specific heat coefficient γ, the larger the value of α in the paramagnetic region.