You can certainly use a steady shear viscosity measurement while mixing the sample. The viscosity of the polymer solution will increase as the sample dissolves and then plateau when fully dissolved, giving a very clear end point to the dissolution time... We've done this very test with a rotational rheometer at Malvern Panalytical with PMMA and MMA monomer and it worked well. using a rheometer has the benefit that the temperature and shear rate are accurately controlled throughout the process and the viscosity is measured precisely. You can even have the test stop automatically when the viscosity changes less than a certain percentage over a specified time period.
I think you should be able to make a mixture of polymer/solvent over the maximum solubility, mix thoroughly and allow the two phases to settle. Then pipet out a known volume of solvent and thoroughly dry and weigh the dissolved polymer.
to use the technique proposed by Dustin Janes, you have to stir the solution for at least, say, 24 hours and preferably under controlled temperature, to assure the dissolution.