Acousto-optic interactions involving propagating phonons can break the time-reversal and frequency-modulation symmetry of light. However, conventional acousto-optic modulators based on bulk materials have frequency bandwidth limited to hundreds of megahertz due to their large structural sizes. Here, we experimentally demonstrate gigahertz single-sideband acousto-optic modulation on an etchless lithium niobate integrated platform by using photonic bound states in the continuum. The upper- or lower-sideband modulation of light can be obtained conveniently by choosing specific combinations of input and output channels. Under this scheme, we have realized a 3-GHz frequency shifter, which operates in the C-band with a 3-dB bandwidth of ~35 nm. The extinction ratios of the upper(lower)-sideband modulated light to the lower(upper)-sideband modulated and unmodulated light are >44 (47) and 25 (23) dB in the 3-dB operating bandwidth. The frequency-shifted light can be further processed with amplitude and frequency modulation. Therefore, the demonstrated gigahertz single-sideband acousto-optic modulation can enable many photonic applications such as optical signal processing, sensing, and ion trapping.