The temperature dependence of the London penetration depth, lambda(T), was measured in both single crystal and polycrystalline MgB2 samples by a high-resolution, radio frequency technique. A clear exponential temperature dependence of lambda(T) was observed at low temperature, indicating s-wave pairing. A BCS fit to the lowest temperature data gives an in-plane energy gap Delta of 30+/-2 K (2Delta/T(c) = 1.5+/-0.1), which is significantly smaller than the standard BCS weak coupling value of 3.5. We find that the data are best described by a two-gap model.