In elastic collision, if $u_1$ and $u_2$ are initial velocities and $v_1$ and $v_2$ are final velocities of bodies with masses $m_1$ and $m_2$ respectively, then

$v_1=\left(\frac{m_1-m_2}{m_1+m_2}\right)u_1+\left(\frac{2m_2}{m_1+m_2}\right)u_2.$

If the second ball is at rest, i.e., $u_2=0,$ then

$v_1=\left(\frac{m_1-m_2}{m_1+m_2}\right)u_1$

According to the question,

$\frac{2}{3}{u}_1=\left(\frac{m_1-m_2}{m_1+m_2}\right)u_1$ $\left\{\because v_1=\frac{2}{3}{u}_1\right\}$

$\Rightarrow 2m_1+2m_2=3m_1-3m_2$

$\Rightarrow -m_1=-5m_2$

$\Rightarrow \frac{m_1}{m_2}=\frac{5}{1}$