jmc

Multiplying both sides by $3x$ gives $9+x^2=3bx$, so $x^2-3bx+9=0$. The equation has exactly one solution if and only if the discriminant of $x^2-3bx+9$ is 0. The discriminant of this quadratic is $(-3b{)}^2-4(9)=9b^2-36$. Setting this equal to 0 gives $9b^2=36$, so $b^2=4$. The positive solution to this equation is $b=\boxed{2}$.