Let \(D\) be a bounded convex domain in the complex plane \(\mathbb{C}\). The support function of \(D\) is defined by \[ h(\theta)=\max_{z\in\overline D}\,\text{ Re}\,(ze^{i\theta}),\qquad\theta\in[0,2\pi]. \] Define \[ \Delta(\theta)=h(\theta)+\int_0^\theta h(t)\,dt,\qquad\theta\in[0,2\pi]. \] Let \(B_2(D)\) be the Bergman space of \(D\). If \(S\) is a continuous linear functional on \(B_2(D)\), then the function \[ \widehat S(\lambda)=S(e^{\lambda z}),\qquad \lambda\in \mathbb{C}, \] is called the Laplace transform of \(S\). The main result of the paper identifies the Laplace transforms of bounded linear functionals on \(B_2(D)\) as entire functions \(F\) satisfying the condition \[ \int_0^{2\pi}\int_0^\infty{| F(re^{i\theta})| ^2\over K(re^{i\theta})}\,dr\,d\Delta(\theta) <\infty, \] where \[ K(\lambda)=\int_D| e^{\lambda z}| ^2\,dA(z). \]