Since the advent of atomic force microscopy(1), mechanical resonators have been used to study a wide variety of phenomena, including the dynamics of individual electron spins(2), persistent currents in normal metal rings(3) and the Casimir force(4,5). Key to these experiments is the ability to measure weak forces. Here, we report on force sensing experiments with a sensitivity of 12 zN Hz(-1/2) at a temperature of 1.2 K using a resonator made of a carbon nanotube. An ultrasensitive method based on cross-correlated electrical noise measurements, in combination with parametric downconversion, is used to detect the low-amplitude vibrations of the nanotube induced by weak forces. The force sensitivity is quantified by applying a known capacitive force. This detection method also allows us to measure the Brownian vibrations of the nanotube down to cryogenic temperatures. Force sensing with nanotube resonators offers new opportunities for detecting and manipulating individual nuclear spins as well as for magnetometry measurements.