Fifth-generation (5G) systems will extensively employ radio access network (RAN) softwarization. This key innovation enables the instantiation of virtual cellular networks running on different slices of the shared physical infrastructure. In this paper, we propose the concept of Private Cellular Connectivity as a Service (PCCaaS), where infrastructure providers deploy covert network slices known only to a subset of users. We then present SteaLTE as the first realization of a PCCaaS-enabling system for cellular networks. At its core, SteaLTE utilizes wireless steganography to disguise data as noise to adversarial receivers. Differently from previous work, however, it takes a full-stack approach to steganography, contributing an LTE-compliant steganographic protocol stack for PCCaaS-based communications, and packet schedulers and operations to embed covert data streams on top of traditional cellular traffic (primary traffic). SteaLTE balances undetectability and performance by mimicking channel impairments so that covert data waveforms are almost indistinguishable from noise. We evaluate the performance of SteaLTE on an indoor LTE-compliant testbed under different traffic profiles, distance and mobility patterns. We further test it on the outdoor PAWR POWDER platform over long-range cellular links. Results show that in most experiments SteaLTE imposes little loss of primary traffic throughput in presence of covert data transmissions (< 6%), making it suitable for undetectable PCCaaS networking.