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In this paper, we propose a new multiple-input- multiple-output (MIMO) transceiver for time-hopping ultra- wideband (TH-UWB) communications. We consider the problem of space-time (ST) coding with binary pulse position modulation (PPM) and we propose the first known family of rate-1 ST codes that can be associated with binary PPM without introducing any additional constellation extension. We prove that the proposed encoding scheme can achieve a full transmit diversity order with 2k transmit antennas. At the receiver side, we propose a maximum-likelihood (ML) decoder that is adapted to the structure of the considered multi-dimensional constellation that does not have the structure of a lattice.