Superconductivity in FeSe has been investigated under high pressure through the measurements of DC magnetization by using a diamond anvil cell. It has been found that superconducting volume fraction abruptly decreases above $\sim$7 GPa ($\sim$5 GPa), when Ar (glycerin) is used as the pressure transmitting media, indicating the disappearance of the superconductivity. The results agree with the appearance of the non-superconducting ortho I\hspace{-1pt}I phase at high pressure observed previously. In contrast, it has been found that the superconductivity survives under pressure even above 7 GPa, consistent with the results previously obtained under hydrostatic pressure by using a cubic anvil apparatus, when the thickness ($t$) of a platelet single crystal specimen is reduced. It is inferred that the appearance of the ortho I\hspace{-1pt}I phase depends sensitively on whether or not the uniaxial stress along the [001] direction is present.