In the CHORUS experiment, a search for associated charm production has been performed in the sample of 130,000 nu interactions. Four events consistent with associated charm production have been found. In the neutral-current sample, three events with associated charm decay topology have been observed with an estimated background of 0.18 +/- 0.05. The relative rate of the associated charm cross-section in deep inelastic nu interactions, sigma(c (c) over bar nu)/sigma(DIS)(NC) = (3.62(-2.42)(+2.95) (stat) +/- 0.54(syst)) x 10(-3) has been measured. In charged-current nu(mu) interactions, one event has a decay topology which is consistent with associated charm production. Based on one event with the estimated background of 0.18 +/- 0.06, we obtained the upper limit on associated charm production in charged-current interactions at 90% C.L. to be sigma(c (c) over bar mu(-))/sigma(CC) < 9.69 x 10(-4).