A Time Projection Chamber (TPC) is an ideal device for the detection of charged particle tracks in a large volume covering a solid angle of almost $4pi$. The high density of hits on a given particle track facilitates the task of pattern recognition in a high-occupancy environment and in addition provides particle identification by measuring the specific energy loss for each track. For these reasons, TPCs with Multiwire Proportional Chamber (MWPC) amplification have been and are widely used in experiments recording heavy-ion collisions. A significant drawback, however, is the large dead time of the order of 1 ms per event generated by the use of a gating grid, which is mandatory to prevent ions created in the amplification region from drifting back into the drift volume, where they would severely distort the drift path of subsequent tracks. For experiments with higher event rates this concept of a conventional TPC operating with a triggered gating grid can therefore not be applied without a significant loss of data. A continuous readout of the signals is the more appropriate way of operation. This, however, constitutes a change of paradigm with considerable challenges to be met concerning the amplification region, the design and bandwidth of the readout electronics, and the data handling. A mandatory prerequisite for such an operation is a sufficiently good suppression of the ion backflow from the avalanche region, which otherwise limits the tracking and particle identification capabilities of such a detector. Gas Electron Multipliers (GEM) are a promising candidate to combine excellent spatial resolution with an intrinsic suppression of ions. In this paper we describe the design, construction and the commissioning of a large TPC with GEM amplification and without gating grid (GEM-TPC).