Collective construction is the research area in which autonomous multi-robot systems build structures according to user specifications. Here we present a hardware system and high-level control scheme for autonomous construction of 3D structures under conditions of gravity. The hardware comprises a mobile robot and specialized passive blocks; the robot is able to manipulate blocks to build desired structures, and can maneuver on these structures as well as in unstructured environments. We describe and evaluate the robot's key capabilities of climbing, navigation, and manipulation, and demonstrate its ability to perform complex tasks that combine these capabilities by having it autonomously build a ten-block staircase taller than itself. In addition, we outline a simple decentralized control algorithm by which multiple simultaneously active robots could autonomously build user-specified structures, working from a high-level description as input.