1. **State the problem:** Find the slope of the line passing through the points $(0,1)$ and $(3,-1)$.\n\n2. **Formula for slope:** The slope $m$ of a line through points $(x_1,y_1)$ and $(x_2,y_2)$ is given by:\n$$m=\frac{y_2 - y_1}{x_2 - x_1}$$\n\n3. **Substitute the points:** Here, $(x_1,y_1) = (0,1)$ and $(x_2,y_2) = (3,-1)$, so:\n$$m=\frac{-1 - 1}{3 - 0}$$\n\n4. **Simplify numerator and denominator:**\n$$m=\frac{-2}{3}$$\n\n5. **Final answer:** The slope of the line is $\boxed{-\frac{2}{3}}$. This means the line falls 2 units vertically for every 3 units it moves horizontally to the right.