A continuous wave (cw) tunable single mode laser beam was propagated perpendicular to a collimated effusive molecular beam of NaI, and the excitation spectra of the A0(+) (upsilon('),J('))-X (1)Sigma(+)(upsilon ",J ") transitions of the NaI molecule were measured by monitoring the intensity of the molecular fluorescence. A large variation of the linewidth and line intensity with the rotational quantum number J was observed. The linewidths of rotationally resolved lines, attributable to lifetime broadening originating from predissociation, were obtained by a fitting of the observed spectral line shape to a Voigt profile.