The precision of global navigation satellite systems (GNSSs) relies heavily on accurate carrier phase ambiguity resolution. The ambiguities are known to take integer values, but the set of ambiguity values is unbounded. We propose a mixture Kalman filter solution to GNSS ambiguity resolution. By marginalizing out the set of ambiguities and exploiting a likelihood proposal for generating the ambiguities, we can bound the possible values to a tight and dense set of integers, which allows for extracting the integer solution as a maximum likelihood estimate from a mixture Kalman filter. We verify the efficacy of the approach in simulation including a comparison with a well-known integer least-squares based method. The results indicate that our proposed switched mixture Kalman filter repeatedly finds the correct integers in cases where the other method fails.