Saying "The distribution of the normal likelihood" is wrong, the normal likelihood is not distributed, what is distributed is $X$ and $\mu$. What you can say is that the posterior has the same form as the likelihood, up to a proportionality constant. Specifically, applying Bayes formula using a uniform prior on $\mu$ between $a$ and $b$ you get $p(\mu|X_i=x,\sigma) = N(x|\mu,\sigma) / \int_a^bN(x|\mu^\prime,\sigma)d\mu^\prime$. Using the limits $a=0$ and $b\to\infty$ that you need, the integral is $\int_0^\infty N(x|\mu^\prime,\sigma)d\mu^\prime = \frac{1}{2}(\textrm{erf}(\frac{x}{\sqrt{2}\sigma})+1)$. Normally, you don't even need to calculate that integral since it doesn't deppend on $\mu$. You can simply compute the integral of $p(\mu|X_i=x,\sigma)$ numerically and use it to normalize the distribution. Also, for estimating the MAP solution for $\mu$ (which in this case is also the maximum likelihood solution due to the uniform prior) you don't need to know the normalization.