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Add $(6/2{)}^2$ and $(-12/2{)}^2$ to both sides to find that the given equation is equivalent to $$(x^2+6x+9)+(y^2-12y+36)=49.$$ The two trinomials on the left-hand side may be rewritten to give $(x+3{)}^2+(y-6{)}^2=7^2$. The set of points $(x,y)$ which satisfy this equation are a distance of 7 units from $(-3,6)$, by the Pythagorean theorem. Hence, the equation defines a circle of radius $7$, which implies that the circumference of the circle is $2\pi \cdot 7=\boxed{14\pi }$.