A complete set of spin-transfer coefficients for forward-angle pp elastic scattering at 198 MeV has been measured at the Indiana University Cyclotron Facility, with uncertainties far smaller than previous measurements. These particular observables, and this kinematic regime, were specifically chosen to maximize the sensitivity of these measurements to the neutral pion-nucleon coupling constant, a fundamental quantity in low and intermediate energy nuclear physics whose value remains highly controversial. To achieve the desired experimental uncertainty, several components of the apparatus required calibration at a higher level of precision than had been previously achieved, and new procedures in data acquisition and analysis were developed in order to minimize many of the possible sources of systematic error. Our primary conclusion from this work is that our results provide strong support for modern potential models of the NN interaction in which a relatively weak pion coupling is employed (g02 ≈ 13.6). Moreover, our data deviate significantly from the predictions of older potentials in which g02 is ~14.4. Calculations carried out in a one-boson exchange framework, using the Bonn B potential, suggest that most of these differences can be accounted for by simply reducing the πNN coupling from 14.4 to 13.6 for the long-range (higher partial wave) pion contributions in these models.