As R W says, the answer is in Plante's paper, but the special case of $1$-dimensional foliations is actually simple, and well-known: let $M$ be the manifold, $*\in M$ be a basepoint, $(\phi^t)$ be the flow (assumed tangential to the boundary of $M$, if any). For every integer $n\ge 0$, let $\mu_n$ be the image in $M$ of the probability measure $dt/(2n)$ on the interval $[-n,+n]$ under the map
$$F:R\to M:t\mapsto\phi^t(*)$$
Since the space of Borelian probability measures on $M$ is compact for the weak topology, the sequence $(\mu_n)$ has a subsequence weakly converging to a Borelian probability measure $\mu$ on $M$.
Claim: $\mu$ is invariant by every $\phi^t$. Indeed, for a fixed $t$, the sequence of measures $$\phi^t_*\mu_n-\mu_n$$ goes to $0$ in the norm topology, since for any continuous real function $f$ on $M$, one has:
$$\vert(\phi^t_*\mu_n-\mu_n)f\vert=\frac{1}{2n}\vert\int_{-n+t}^{n+t}f(\phi^s(*))ds-\int_{-n}^{+n}f(\phi^s(*))ds\vert\le\frac{t}{n}\Vert f\Vert_\infty$$
Finally, the $R$-invariant measure $\mu$ amounts to a transverse invariant measure $\nu$ such that locally (in every local flow box $B\cong D^{n-1}\times I$) one has $\mu=\nu\otimes dt$. The measure $\nu(D)$ of every small transverse disk $D$ is the limit of some *subsequence* of the sequence
$$\frac{1}{2n}{\sharp(F^{-1}(D)\cap[-n,+n])}$$
In this construction, we have used the choice axiom (hidden in the compacity of the measures space); in particular, the construction of the subsequence is non-constructive.