This work provides tight upper- and lower-bounds for the problem of mean estimation under $ε$-differential privacy in the local model, when the input is composed of $n$ i.i.d. drawn samples from a normal distribution with variance $σ$. Our algorithms result in a $(1-β)$-confidence interval for the underlying distribution's mean $μ$ of length $\tilde O\left( \frac{σ\sqrt{\log(\frac 1 β)}}{ε\sqrt n} \right)$. In addition, our algorithms leverage binary search using local differential privacy for quantile estimation, a result which may be of separate interest. Moreover, we prove a matching lower-bound (up to poly-log factors), showing that any one-shot (each individual is presented with a single query) local differentially private algorithm must return an interval of length $Ω\left( \frac{σ\sqrt{\log(1/β)}}{ε\sqrt{n}}\right)$.