The NEMO 3 detector, which is devoted to searching for neutrinoless double beta decay (0νββ), has been operating in the Fréjus Underground Laboratory in France since February 2003. Half-lives of the two-neutrino double beta decays (2νββ) have been measured for ¹⁰⁰Mo and ⁸²Se. After 389 days of data collection from February 2003 until September 2004, no evidence for neutrinoless double beta decay was found from 7 kg of ¹⁰⁰Mo and 1 kg of ⁸²Se. The corresponding lower limits for the half-lives are 4.6 × 10²³ y at 90% C.L. for ¹⁰⁰Mo and 1.0 × 10²³ y at 90% C.L. for ⁸²Se. Limits for the effective Majorana neutrino mass are 〈m_ν〉 < 0.66 – 2.81 eV for ¹⁰⁰Mo and 〈m_ν〉 < 1.75 – 4.86 eV for ⁸²Se. Radon was the dominant background and is now strongly suppressed by a radon-trapping facility, which has been in operation since October 2004. The expected sensitivity of the NEMO 3 experiment after five years of data collection will be T_1/2^0νββ > 2 × 10²⁴ y at 90% C.L. for ¹⁰⁰Mo and 8.3 × 10²³ y at 90% C.L. for ⁸²Se, corresponding to 〈m_ν〉 < 0.3 – 1.3 eV and 〈m_ν〉 < 0.6 – 1.7 eV respectively.