We report the generation of 3.3-mW single-cycle terahertz (THz) pulses at 1-kHz repetition rate via optical rectification in MgO-doped prism-cut stoichiometric LiNbO3. Efficient pulse-front tilting of 800-nm pulses was realized by an optimized single-lens focusing scheme for radially-symmetric propagation of THz beams. In this geometry, nearly-diffraction-limited THz Gaussian beams with electric field strength as high as 350 kV/cm were generated. The pump-to-THz energy conversion efficiency of $1.36{ imes}10^{-3}$ and the extremely high signal-to-noise ratio of ~1:15000 achieved are among the best results for 1-kHz single-cycle terahertz pulse generation ever demonstrated in room temperature operation.