We consider two jointly stationary and ergodic random measures $ξ$ and $η$ on the real line $\mathbb{R}$ with equal intensities. An allocation is an equivariant random mapping from $\mathbb{R}$ to $\mathbb{R}$. We give sufficient and partially necessary conditions for the existence of allocations transporting $ξ$ to $η$. An important ingredient of our approach is to introduce a transport kernel balancing $ξ$ and $η$, provided these random measures are mutually singular. In the second part of the paper, we apply this result to the path decomposition of a two-sided Brownian motion into three independent pieces: a time reversed Brownian motion on $(-\infty,0]$, an excursion distributed according to a conditional Itô's law and a Brownian motion starting after this excursion. An analogous result holds for Bismut's excursion law.