Two-proton decay of the 17 Ne low-lying states populated in the 1 H( 18 Ne, d) 17 Ne transfer reaction was studied. The two-proton width Γ2p of the 17 Ne first excited 3/2 − state at E * = 1.288 MeV is of importance for the two-proton radioactivity theory and nuclear-astrophysics applications. A dedicated search for the two-proton emission of this state was performed leading to the new upper limit obtained for the width ratio Γ2p/Γγ < 1.6(3) × 10 −4 . A novel, "combined mass" method is suggested and tested capable to improve the resolution of the experiment which is a prime significance for the study of nuclear states with extremely small particle-to-gamma width ratios Γpart/Γγ . The condition Γpart ≪ Γγ is quite common for the states of astrophysical interest which makes the proposed approach promising in this field.