Robotic manipulation of cloth has applications ranging from fabrics manufacturing to handling blankets and laundry. Cloth manipulation is challenging for robots largely due to their high degrees of freedom, complex dynamics, and severe self-occlusions when in folded or crumpled configurations. Prior work on robotic manipulation of cloth relies primarily on vision sensors alone, which may pose challenges for fine-grained manipulation tasks such as grasping a desired number of cloth layers from a stack of cloth. In this paper, we propose to use tactile sensing for cloth manipulation; we attach a tactile sensor (ReSkin) to one of the two fingertips of a Franka robot and train a classifier to determine whether the robot is grasping a specific number of cloth layers. During test-time experiments, the robot uses this classifier as part of its policy to grasp one or two cloth layers using tactile feedback to determine suitable grasping points. Experimental results over 180 physical trials suggest that the proposed method outperforms baselines that do not use tactile feedback and has better generalization to unseen cloth compared to methods that use image classifiers. Code, data, and videos are available at https://sites.google.com/view/reskin-cloth.