  Liouville quantum gravity (LQG) and the Brownian map (TBM) are two distinct
models of measure-endowed random surfaces. LQG is defined in terms of a real
parameter $\gamma$, and it has long been believed that when $\gamma =
\sqrt{8/3}$, the LQG sphere should be equivalent (in some sense) to TBM.
However, the LQG sphere comes equipped with a conformal structure, and TBM
comes equipped with a metric space structure, and endowing either one with the
other's structure has been an open problem for some time.
  This paper is the first in a three-part series that unifies LQG and TBM by
endowing each object with the other's structure and showing that the resulting
laws agree. The present work uses a form of the quantum Loewner evolution (QLE)
to construct a metric on a dense subset of a $\sqrt{8/3}$-LQG sphere and to
establish certain facts about the law of this metric, which are in agreement
with similar facts known for TBM. The subsequent papers will show that this
metric extends uniquely and continuously to the entire $\sqrt{8/3}$-LQG surface
and that the resulting measure-endowed metric space is TBM.
