  Let $E$ be an elliptic curve over $\mathbb{Q}$ and $\varrho_1, \varrho_2
\colon \mathrm{Gal}(H/\mathbb{Q}) \to \mathrm{GL}_2(L)$ be two odd Artin
representations. We use $p$-adic methods to investigate the part of the
Mordell-Weil group $E(H) \otimes L$ on which the Galois group acts via
$\varrho_1 \otimes \varrho_2$. When the rank of the group is two,
Darmon-Lauder-Rotger used a dominant triple product $p$-adic $L$-function to
study this group, and gave an Elliptic Stark Conjecture which relates its value
outside of the interpolation range to two Stark points and one Stark unit. Our
paper achieves a similar goal in the rank one setting. We first generalize
Hsieh's construction of a 3-variable balanced triple product $p$-adic
$L$-function in order to allow Hida families with classical weight one
specializations. We then give an Elliptic Stark Conjecture relating its value
outside of the interpolation range to a Stark point and two Stark units. As a
consequence, we give an explicit $p$-adic formula for a point which should
conjecturally lie in $E(H) \otimes L$. We prove our conjecture for dihedral
representations associated with the same imaginary quadratic field. This
requires a generalization of the results of Bertolini-Darmon-Prasanna which we
prove in the appendix.
