The answer is yes. 

First, let me point out that in general, in ZFC we are not able to refer to the notion of *first-order-definable-in-$L$*, since definability is not expressible. But in your case, we have $0^\#$, from which we are able to define a truth predicate for first-order truth in $L$, and so your question can be formulated.

For each natural number $n$, the least $\Sigma_n$-correct cardinal $\kappa_n$ is definable in $L$, meaning $L_{\kappa_n}\prec_{\Sigma_n}L$, since we can express this property using a $\Sigma_n$ truth predicate. Further, any $\Sigma_n$ definable ordinal will be bounded by $\kappa_n$. Thus, $\kappa=\sup_n\kappa_n$. 

But the union of an increasingly elementary chain is elementary, so $L_\kappa\prec L$ and thus $\kappa$ is a Silver indiscernible. 

This answers your question, but let me augment my answer with the following observation for the general ZFC case, which I find interesting.

**Theorem.** For any model $M$ of ZFC, not necessarily well-founded, let $W$ be the collection of all $x\in M$ for which $x\in (V_\alpha)^M$ where $\alpha$ is an ordinal definable in $M$ without parameters. This is called the *definable cut* of $M$. Then $W\prec M$. 

**Proof.** Notice that the definable cut may not necessarily have a least upper bound in $M$. It could be that $W=M$ or that the supremum of $W$ is not realized in $M$. 

But we can nevertheless prove that $W$ is an elementary substructure of $M$ by verifying the Tarski-Vaught criterion. If $M$ has a witness for an extensional statement with parameters in $W$, then the least rank of such a witness is definable from those parameters, and by considering all possible parameters up to a definable rank, we can get a definable bound on the rank of the witnesses. And so the witness is in $W$. So we have fulfilled the Tarski-Vaught criterion, and thus $W\prec M$. $\quad\Box$

Note that in this theorem, we are using the external notion of definability, coming from outside the model, whereas in your question, we were using the internal notion of definability provided by the truth predicate defined fro $0^\#$. These are not necessarily the same, even when $0^\#$ exists, since there could be nonstandard models of ZFC with $0^\#$. this issue can be seen as a possible ambiguity in your question, as to whether you intend to use the internal notion of definability or the meta-theoretic notion of definability. In the latter case, the supremum of the meta-theoretically definable ordinals will not exist in the model, since from it you could define the standard cut, which is impossible.