One may simply use the theorem of Boltzmann's that is in the very Wikipedia article you point to.
Note that specifying the mean and variance is equivalent to specifying the first two raw moments - each determines the other (it's not actually necessary to invoke this, since we may apply the theorem directly to the mean and variance, it's just a little simpler this way).
The theorem then establishes that the density must be of the form:
$$f(x)=c \exp\left(\lambda_1 x + \lambda_2 x^2 \right)\quad \mbox{ for all } x > 0$$
Integrability over the positive real line will restrict $\lambda_2$ to be $\leq 0$.
To my surprise (since I wouldn't have expected it when I started this answer), this appears to leave us with a truncated normal distribution.