Selective presaturation is a common technique for suppressing excessive solvent signals during proton NMR analysis of dilute samples in protic solvents. When the solvent T1 relaxation time constant varies within a series of samples, parameters for the presaturation sequence must often be re-adjusted for each sample. The EXCEPT (EXponentially Converging Eradication Pulse Train) presaturation pulse sequence was developed to eliminate time consuming pulse-parameter re-optimization as long as the variation in the solvent's T1 remains within an order of magnitude. EXCEPT consists of frequency-selective inversion pulses with progressively decreasing interpulse delays. The interpulse delays were optimized to encompass T1 relaxation times ranging from 1 to 10s, but they can be easily adjusted by a single factor for other ranges that fall within an order of magnitude with respect to T1. Sequences with different numbers of inversion pulses were tested to maximize suppression while minimizing the number of pulses and thus the total time needed for suppression. The EXCEPT-16 experiment, where 16 denotes the number of inversion pulses, was found satisfactory for many standard applications. Experimental results demonstrate that EXCEPT provides effective T1-insensitive solvent suppression as predicted by the theory. The robustness of EXCEPT with respect to changes in solvent T1 allows NMR investigations to be carried out for a series of samples without the need for pulse-parameter re-optimization for each sample.