We propose a new first-order augmented Lagrangian algorithm ALCC to solve conic convex programs of the form $\min \{r(x) + g(x) : Ax-b \in K; x \in Q\}$; where $r: R^n \rightarrow R\cup{+\infty}$ and $g:R^n\rightarrow R$ are closed, convex functions and $g$ has a Lipschitz-continuous gradient; $A$ is an $m\times n$ matrix, $K$ is a closed convex cone, and $Q$ is a subset of $\mathrm{dom}(r)$ such that $Q$ is a ``simple" convex compact set in the sense that optimization problems of the form $\min\{r(x) + \|x-x_0\|_2^2: x \in Q\}$ can be efficiently solved. We show that any limit point of the primal ALCC iterate sequence is an optimal solution of the conic convex problem, and the dual ALCC iterate sequence has a unique limit point which is a KKT point of the conic problem. We also show that for all $\varepsilon> 0$, the primal ALCC iterates are $\varepsilon$-feasible and $\varepsilon$-optimal after $O(\log(1/\varepsilon))$ iterations, which require $O(1/\varepsilon \log(1/\varepsilon))$ oracle calls to solve problems of the form $\min\{r(x) + \|x-x_0\|_2^2: x \in Q\}$.