The hyperthermophilic archaeon Pyrodictium grows in the form of a macroscopically visible network. It consists of cells entrapped in an extracellular matrix of hollow tubules, the "cannulae." Here, we present the three-dimensional structure of a single cell in conjunction with two extracellular cannulae, as determined by cryo-electron microscopy. To achieve this, the information from two independent tilt series of the same specimen was combined, with the specimen rotated in the second series. In the three-dimensional tomographic reconstruction, we were able to trace the two cannulae in their full length, in particular, also inside the cell. One cannula enters the periplasmic space, while the other cannula contacts the surface of the cell, the S-layer. This indicates that the cannulae interconnect individual cells with each other on the level of their periplasmic space; we do not, however, have evidence that they enter the cytoplasm of the cells. The implications of these data for possible functions of the cannulae are discussed.