High resolution spectrum of the B 1Π–X 1Σ+ transition of the NaRb molecule was measured with the technique of the Doppler-free laser polarization spectroscopy. Molecular constants of the B 1Π(v=0–12) and X 1Σ+(v=0–6) states of 23Na85Rb were determined. The energy levels of the B 1Π state were found to present many irregularities due to perturbations. The resonance fluorescence spectrum following an excitation to a strongly perturbed level was measured. The fluorescence to the (1)3Σ+ state, which consists of discrete lines followed by a continuum band, was observed in addition to the fluorescence lines to rovibrational levels of the X 1Σ+ state. The perturbing state to the B 1Π(v=8,J=15–21) levels is identified as the (1)3Π state by comparing the observed fluorescence spectra with the selection rules for perturbations and radiative transitions. The dissociation limit of the (1)3Σ+ state, which separates into the Na(3s2S1/2)+Rb(5s2S1/2) atoms, was deduced from the spectrum. The dissociation energies of the X 1Σ+, (1)3Σ+, and B 1Π states were determined to be 5030±2, 182±2, and 1319±2 cm−1, respectively.