For a class of uncertain nonlinear systems with sector-bounded nonlinearity, a proportional-integral state observer is designed in a dissipativity framework. The dissipativity ensures the bounded estimation errors against the bounded disturbance and an integral term in the observer structure can provide additional degrees of freedom in the observer design that can be used to improve the estimation performance. In this study, a dissipative condition for the proposed observer is found in terms of a linear matrix inequality (LMI). In order to specify the dissipativity, the L2 gain between the disturbance and the weighted sum of estimation errors is adopted as the supply rate in the dissipativity framework. In addition, the notion of the proposed design method is extended to the exponential dissipativity. The effectiveness of the proposed observer is demonstrated through a numerical example.