An analysis of about 27 h of fast photometry of the cool ZZ Ceti white dwarf PG2303+243 reveals (1) the presence of many modes (as many as 22), some of them of low enough amplitude to have escaped previous detection, and (2) a short time scale (less than 10 d) drastic variation in amplitude. A preliminary identification of the modes observed in PG2303+243 is carried out on the basis of the assumptions that the frequency spectrum is stable and that it can be represented by the theoretical spectrum deduced from the linear adiabatic analysis. It is suggested that 10 of the 22 modes observed may be identified as g-modes of l = 1, 2, 3, and 4, trapped, or partially trapped, in the outer hydrogen envelope. According to the present analysis, the mass of the hydrogen envelope in PG2303+243 would be constrained in the range of -8.7 to -7.6, if a total mass of 0.6 solar mass and an effective temperature of 12,000 K are assumed.