  We prove that there exist Salem numbers with trace $-3$ and every even degree
$\geq 34$. Our proof combines a theoretical approach, which allows us to treat
all sufficiently large degrees, with a numerical search for small degrees.
Since it is known that there are no Salem numbers of trace $-3$ and degree
$\leq 30$, our result is optimal up to possibly the single value $32$, for
which it is expected there are no such numbers.
