1. **State the problem:** Simplify the expression $$\frac{1B^3}{-3a^3c^3} \cdot \frac{a^2B}{-4B^2}$$. 2. **Write the expression as a single fraction:** $$\frac{1B^3}{-3a^3c^3} \times \frac{a^2B}{-4B^2} = \frac{1B^3 \cdot a^2B}{-3a^3c^3 \cdot -4B^2}$$ 3. **Multiply numerators and denominators:** $$\frac{B^3 \cdot a^2 \cdot B}{(-3)(-4) a^3 c^3 B^2} = \frac{a^2 B^{3+1}}{12 a^3 c^3 B^2} = \frac{a^2 B^4}{12 a^3 c^3 B^2}$$ 4. **Simplify powers of variables:** $$= \frac{\cancel{a^2} a^{\cancel{2}} B^{\cancel{2}} B^{2}}{12 \cancel{a^3} a^{\cancel{2}} c^3 \cancel{B^2}} = \frac{B^{4-2}}{12 a^{3-2} c^3} = \frac{B^2}{12 a c^3}$$ 5. **Final simplified expression:** $$\boxed{\frac{B^2}{12 a c^3}}$$ This means the original expression simplifies to $$\frac{B^2}{12 a c^3}$$ after canceling common factors and combining powers.