We report the first measurements of the perturbation amplitude in the converging Richtmyer-Meshkov instability in a semiannular shock tube. At early stages, the amplitude growth agrees well with the impulsive model considering the geometrical convergence effect. A quick decrease of the growth rate at late time, even to be negative, before the reshock is observed for the first time. The reduction of the growth rate is ascribed to the Rayleigh-Taylor stabilization caused by the interface deceleration motion only presented in the converging circumstance. By reasonably evaluating the Rayleigh-Taylor stabilization, a modified model based on the Bell equation is proposed, which well predicts the perturbation growth in a converging geometry from early to late stages before the reshock. It is also found that the flow compressibility is significant in the converging Richtmyer-Meshkov instability.