The next wave driving the expansion of the Internet will come from the Internet of Things. Bluetooth LE is a rapidly emerging ultra-low-power radio technology expected to be incorporated in billions of IoT devices in the next few years. Consequently, it is particularly important to specify Internet connectivity solutions for Bluetooth LE. In this article we present such solutions based on the ongoing standardization work in the IETF and Bluetooth Special Interest Group. We prove the feasibility of a complete IP-based protocol stack on constrained devices and illustrate its performance, highlighting key trade-offs. In addition, we discuss gateway operation covering global IPv6 connectivity and proxy-cache functionality.