  The rank one Gross conjecture for Deligne-Ribet $p$-adic $L$-functions was
solved in works of Darmon-Dasgupta-Pollack and Ventullo by the Eisenstein
congruence among Hilbert modular forms. The purpose of this paper is to prove
an analogue of the Gross conjecture for the Katz $p$-adic $L$-functions
attached to imaginary quadratic fields via the congruences between CM forms and
non-CM forms. The new ingredient is to apply the $p$-adic Rankin-Selberg method
to construct a non-CM Hida family which is congruent to a Hida family of CM
forms at the $1+\varepsilon$ specialization.
