  The regular Dirichlet extension is the dual concept of regular Dirichlet
subspace. The main purpose of this paper is to characterize all the regular
Dirichlet extensions of one-dimensional Brownian motion and to explore their
structures. It is shown that every regular Dirichlet extension of
one-dimensional Brownian motion may essentially decomposed into at most
countable disjoint invariant intervals and an $\mathcal{E}$-polar set relative
to this regular Dirichlet extension. On each invariant interval the regular
Dirichlet extension is characterized uniquely by a scale function in a given
class. To explore the structure of regular Dirichlet extension we apply the
idea introduced in [17], we formulate the trace Dirichlet forms and attain the
darning process associated with the restriction to each invariant interval of
the orthogonal complement of $H^1_\mathrm{e}(\mathbb{R})$ in the extended
Dirichlet space of the regular Dirichlet extension. As a result, we find an
answer to a long-standing problem whether a pure jump Dirichlet form has proper
regular Dirichlet subspaces.
