We analyze superconductivity in MgB₂ observed below T_c = 39 K resulting from electron-phonon coupling involving a mode at ℏω₁ = 24 meV and, most importantly, the in-plane B-B E_2g vibration at ℏω₂ = 67 meV. The quasiparticles originating from π and σ states couple strongly to the low-frequency mode and the E_2g vibrations, respectively. Using two-band Eliashberg theory, λ_π = 1.4 and λ_σ = 0.7, we calculate the gap functions Δⁱ(ω,O) (i = π, σ). Our results provide an explanation of recent tunneling experiments. We get H_c2^ab/H_c2^c ≈ 3.9.