  The main purpose of this paper is to find double blocking sets in
$\mathrm{PG}(2,q)$ of size less than $3q$, in particular when $q$ is prime. To
this end, we study double blocking sets in $\mathrm{PG}(2,q)$ of size $3q-1$
admitting at least two $(q-1)$-secants. We derive some structural properties of
these and show that they cannot have three $(q-1)$-secants. This yields that
one cannot remove six points from a triangle, a double blocking set of size
$3q$, and add five new points so that the resulting set is also a double
blocking set. Furthermore, we give constructions of minimal double blocking
sets of size $3q-1$ in $\mathrm{PG}(2,q)$ for $q=13$, $16$, $19$, $25$, $27$,
$31$, $37$ and $43$. If $q>13$ is a prime, these are the first examples of
double blocking sets of size less than $3q$. These results resolve two
conjectures of Raymond Hill from 1984.
