We develop a learning-based algorithm for the control of autonomous systems governed by unknown, nonlinear dynamics to satisfy user-specified spatio-temporal tasks expressed as signal temporal logic specifications. Most existing algorithms either assume certain parametric forms for the unknown dynamic terms or resort to unnecessarily large control inputs in order to provide theoretical guarantees. The proposed algorithm addresses these drawbacks by integrating neural-network-based learning with adaptive control. More specifically, the algorithm learns a controller, represented as a neural network, using training data that correspond to a collection of system parameters and tasks. These parameters and tasks are derived by varying the nominal parameters and the spatio-temporal constraints of the user-specified task, respectively. It then incorporates this neural network into an online closed-form adaptive control policy in such a way that the resulting behavior satisfies the user-defined task. The proposed algorithm does not use any a priori information on the unknown dynamic terms or any approximation schemes. We provide formal theoretical guarantees on the satisfaction of the task. Numerical experiments on a robotic manipulator and a unicycle robot demonstrate that the proposed algorithm guarantees the satisfaction of 50 user-defined tasks, and outperforms control policies that do not employ online adaptation or the neural-network controller. Finally, we show that the proposed algorithm achieves greater performance than standard reinforcement-learning algorithms in the pendulum benchmarking environment.