A silk hat type flexible external gear (13) of a flexible mesh engagement gear device (11) comprises a barrel portion (22), a diaphragm portion (23) and a boss portion (25). A thin portion (221) whose thickness is 80 % of the thickness of a main body portion of the barrel portion (22) is formed between the main body portion and a curved portion (222). The thickness of the diaphragm portion (23) is set such that it is the maximum at its internal circumferential end portion (point A), the minimum at its central portion (point B) and a slightly thinner than the maximum thickness at an external circumferential end portion (point C). The contour of the diaphragm is determined by using a group of curves (223, 224) such that the change in thickness is smooth as a whole. It is possible to ease the concentration of stress at the internal and external circumferential end portions so as to realize a gentle stress distribution by disposing the thin portion (221) in a way as described above and determining the thickness of the diaphragm portion (23) also in a way as described above. Consequently, it is possible to realize a silk hat type flexible mesh engagement gear device having a shorter axial length and a smaller outside diameter.