Cut, seed, and dice the melon into chunks, discarding the rind. Combine half the melon and the lime juice in a blender and puree until smooth. Pour through a strainer into a large bowl. Puree the remaining melon and 1 cup water and pour through the strainer. Use a wooden spoon to press out as much melon liquid as possible. Discard the melon pulp. Transfer the melon juice to a pitcher.

In a small saucepan, combine the remaining cup of water, sugar, and honey. Bring to a simmer, stirring until the sugar is dissolved. Pour into the pitcher with the melon juice. Chill in the refrigerator until cold. Serve over ice.