Shrinking mechanical resonators to submicrometer dimensions (approximately 100 nm) has tremendously improved capabilities in sensing applications. In this Letter, we go further in size reduction using a 1 nm diameter carbon nanotube as a mechanical resonator for mass sensing. The performances, which are tested by measuring the mass of evaporated chromium atoms, are exceptional. The mass responsivity is measured to be 11 Hz x yg(-1) and the mass resolution is 25 zg at room temperature (1 yg = 10(-24) g and 1 zg = 10(-21) g). By cooling the nanotube down to 5 K in a cryostat, the signal for the detection of mechanical vibrations is improved and corresponds to a resolution of 1.4 zg.