We consider the problem of Bayesian optimization (BO) in one dimension, under a Gaussian process prior and Gaussian sampling noise. We provide a theoretical analysis showing that, under fairly mild technical assumptions on the kernel, the best possible cumulative regret up to time $T$ behaves as $Ω(\sqrt{T})$ and $O(\sqrt{T\log T})$. This gives a tight characterization up to a $\sqrt{\log T}$ factor, and includes the first non-trivial lower bound for noisy BO. Our assumptions are satisfied, for example, by the squared exponential and Matérn-$ν$ kernels, with the latter requiring $ν> 2$. Our results certify the near-optimality of existing bounds (Srinivas {\em et al.}, 2009) for the SE kernel, while proving them to be strictly suboptimal for the Matérn kernel with $ν> 2$.