Most existing space-time (ST) coding schemes have been developed for flat fading channels. To offer antenna diversity gains, they rely on channel state information (CSI) acquired at the receiver through training, or, via blind channel estimation. Differential ST modulation obviates the need for CSI but is less effective in rapidly fading environments. This paper develops novel ST coders and decoders that enjoy transmit-diversity gains in the presence of time-selective fading channels. Modeling the time-selective channels as random processes, we employ Kalman filtering to estimate them and mitigate their effects so that symbol detection benefits from transmit-diversity. Computer simulations confirm that the proposed scheme achieves robust performance in time-selective channels with a few training symbols.