  In this paper, we prove the following asymptotic formula for the spectral
cubic moment of central $L$-values: $$
  \sum_{t_f \leqslant T} \frac {2 L \big( \tfrac 1 2 , f \big)^3} {L(1,
\mathrm{Sym}^2 f)} + \frac {2} {\pi} \int_{0}^{T} \frac {\left| \zeta
\big(\tfrac 1 2 + it \big) \right|^{6} } { | \zeta (1 + 2 it ) |^2 } \mathrm{d}
t = T^2 P_3 (\log T) + O (T^{1+\varepsilon}) , $$ where $f$ ranges in an
orthonormal basis of (even) Hecke--Maass cusp forms, and $P_3$ is a certain
polynomial of degree $3$. It improves on the error term $O
(T^{8/7+\varepsilon})$ in a paper of Ivi\'c and hence confirms his strong
conjecture for the cubic moment. This is the first time that the (strong)
moment conjecture is fully proven in a cubic case. Moreover, we establish the
short-interval variant of the above asymptotic formula on intervals of length
as short as $T^{\varepsilon}$.
