We establish a new concentration-of-measure inequality for the sum of independent random variables with β-heavy tail. This includes exponential of Gaussian distributions (a.k.a. log-normal distributions), or exponential of Weibull distributions, among others. These distributions have finite polynomial moments at any order but many not have finite α-exponential moments. We exhibit a new Orlicz norm adapted to this setting of β-heavy tails, we prove a new Talagrand inequality for the sum and a new maximal inequality. As consequence, a deviation probability of the sum from its mean is obtained.