Neural network with much less positive examples
There are a couple of things you can do to treat the class imbalance. 


A way is to use an SVM-loss (Hinge-loss) function in the network, where the margin around the decision boundary is maximized. The good part of this: Because this function considers the support vectors only, the data points which do not lie close to the margin will not be taken into account. Of course, with more data, you almost guarantee to change the decision boundary and therefore the support vectors. However, especially if your data is spread around the boundary (if you can sample some positive-like true negatives), then you should get close to the correct minimum.
This is an easy way to get around without much effort.
You could augment (not just duplicate) the positives more than you augment the negatives, so that the data becomes balanced. 
The weighting of the samples is already discussed, and I think it is valid.