A dealer professes to sell his goods at cost price but uses a false weight and thus earns $25\%$ profit. For weighing a kilogram, he uses a false weight of ______.

Let the false weight used for $1\text{kg}$ ($1000\text{gm}$) be $x\text{gm}$.The profit percentage when selling at cost price using false weight is given by the formula:
$$\text{Profit Percentage}=\left(\frac{\text{True Weight}-\text{False Weight}}{\text{False Weight}}\times 100\right)\%$$
Given that Profit Percentage = $25\%$ and True Weight = $1000\text{gm}$:
$$25=\frac{1000-x}{x}\times 100$$
$$\frac{25}{100}=\frac{1000-x}{x}$$
$$\frac{1}{4}=\frac{1000-x}{x}$$
$$x=4(1000-x)$$
$$x=4000-4x$$
$$5x=4000$$
$$x=800\text{gm}$$
Hence, the dealer uses a false weight of $800\text{gm}$.