For very low relaxation time (i.e., lesser than a microsecond), viscoelastic fluid experimental determination is difficult, if not impossible. In this work, the relaxation time measurement of a weakly elastic polymer solution, too low to be measured using classical rheometry techniques, is assessed using a mixed experimental-numerical strategy. First the fluid is rheologically assessed, by measuring its shear viscosity, surface tension, and density. Then the relaxation time is determined by comparing the jetting of polymer solution from a continuous ink-jet (CIJ) device, experimentally and numerically. The numerical approach is first validated using test cases, and a viscoelastic Oldroyd-B model is used to model the experimental solution. The relaxation time is then a parameter allowing us to fit numerical simulation onto experimental results. This mixed strategy is particularly convenient for weakly elastic solution for which physical parameters cannot be measured using an experimental rheometry setup.