An option of the American type with the payment function \[ f_t= e^{- \lambda t} \Biggl[ \int^t_0 S_u du+ s\psi_0\Biggr] \] (where \(\lambda> 0\), \(\psi_0\geq 0\) are constants, \(s= S_0\)) is considered in a diffusion model of a BS-market consisting of two assets: a riskless bank account \(B= (B_t)_{t\geq 0}\), and a stock \(S= (S_t)_{t\geq 0}\) described by the relations: \[ B_t= B_0 e^{rt},\;B_0> 0\text{ and } S_t= S_0 e^{\mu t}\exp \Biggl\{\sigma W_t- {\sigma^2\over 2} t\Biggr\}, \] where \(W_t= W_t(\omega)\) is the standard Wiener process [see \textit{A. N. Shiryaev}, \textit{Yu. M. Kabanov}, the first author and \textit{A. V. Mel'nikov}, Teor. Veroyatn. Primen. 39, No. 1, 23-129 (1994)]. We determine the correct value of the considered integral option. We also derive the rational time to present the given option to execution.