We derive the optimal $(0, δ)$-differentially private query-output independent noise-adding mechanism for single real-valued query function under a general cost-minimization framework. Under a mild technical condition, we show that the optimal noise probability distribution is a uniform distribution with a probability mass at the origin. We explicitly derive the optimal noise distribution for general $\ell^p$ cost functions, including $\ell^1$ (for noise magnitude) and $\ell^2$ (for noise power) cost functions, and show that the probability concentration on the origin occurs when $δ> \frac{p}{p+1}$. Our result demonstrates an improvement over the existing Gaussian mechanisms by a factor of two and three for $(0,δ)$-differential privacy in the high privacy regime in the context of minimizing the noise magnitude and noise power, and the gain is more pronounced in the low privacy regime. Our result is consistent with the existing result for $(0,δ)$-differential privacy in the discrete setting, and identifies a probability concentration phenomenon in the continuous setting.