Fourier Transform Spectrometers (FTS) are commonly operated in a rapid-scan (RS) mode, in which an interferogram of an astronomical source is obtained as quickly as possible, followed by one of a nearby background position. In an alternate operating mode, known as step-and-integrate (SI), the optical path difference in the interferometer is incremented in discrete steps, and the signal is integrated only when the interferometer mirrors are stationary. This mode requires some other means of modulating the signal, such as chopping the secondary mirror so that the detector alternately views source and background. The noise bandwidth in the SI mode (typically ~1 Hz) is much smaller than in the RS mode (~1 KHz), which in principle can lead to an increase in overall sensitivity. The main problem with the SI mode is that it takes much longer (~30x) to acquire an interferogram. At submillimetre wavelengths, through the use of narrowband optical filters, which are matched to regions of low atmospheric opacity, it is possible to sample the interferogram at less than the interval determined from the DC band limited Nyquist frequency (a condition known as aliasing) and still unambiguously recover the spectral information. We describe in detail the aliased, SI mode of operation of an FTS at the JCMT and present first results of astronomical spectra obtained using this mode. The resulting spectra are compared and contrasted to data obtained in the RS mode.