Decision Tree with Unbalanced Data
Most common approach to such problems is some form of sub-sampling: randomly draw equal numbers of examples for each class. For model tuning you can use Stratified Cross-validation which in essence uses exactly this approach.

Other possibility is to weight the examples according to the ratio of class sizes. However, since on of your class has infinite number of samples this is impossible, unless you'll first employ sub-sampling anyway, at least for this infinite class. One example for decision tree context is modified information gain: "A Robust Decision Tree Algorithm for Imbalanced Data Sets" by Wei Liu, Sanjay Chawla, David A. Cieslak, Nitesh V. Chawla (PDF)