To classify galaxies morphologically, we developed Galaxy Morphology Network, a convolutional neural network that classifies galaxies according to their bulge-to-total ratio. GaMorNet does not need a large training set of real data and can be applied to datasets with a range of signal-to-noise ratios and spatial resolutions. We first trained GaMorNet on simulations of galaxies with a bulge and a disk component and then used a technique called transfer learning to refine the already trained network using ∼25% of the real dataset to achieve misclassification rates of ≲5%. This has very important consequences, as the applicability of CNNs to future data-intensive surveys like LSST, WFIRST, and Euclid will depend on their ability to perform on multiple data-sets without the need for a large training set of real data. Using the GaMorNet classifications, we study the quenching of star formation in ∼100,000∼100,000 (z∼0z∼0) SDSS and ∼20,000 (z∼1) CANDELS galaxies. We find that bulge- and disk-dominated galaxies have completely different color-mass diagrams, in agreement with previous studies. For both SDSS and CANDELS galaxies, disk-dominated galaxies peak in the blue cloud, across a broad range of masses, consistent with slow exhaustion of star-forming gas with no rapid quenching. A small population of red disks is found at high mass (∼14% of disks at z∼0 and 2% of disks at z∼1). In contrast, bulge-dominated galaxies are mostly red, with much smaller numbers down towards the blue cloud, suggesting rapid quenching and fast evolution across the green valley.