A plane cuts a sphere of radius $ 1$ into two pieces, one of which has three times the surface area of the other. What is the area of the disk that the sphere cuts out of the plane?
1. **Understanding the problem**: We need to find the area of the disk that a plane cuts out of a sphere of radius 1, given that the plane divides the sphere into two pieces with a surface area ratio of 3:1.

2. **Surface area of a sphere**: The total surface area \( A \) of a sphere with radius \( r \) is given by:
   \[
   A = 4 \pi r^2
   \]
   For a sphere of radius 1, the surface area is:
   \[
   A = 4 \pi \cdot 1^2 = 4 \pi
   \]

3. **Surface area ratio**: The plane divides the sphere into two parts with a surface area ratio of 3:1. Let the surface areas of the two parts be \( 3x \) and \( x \). Therefore:
   \[
   3x + x = 4x = 4 \pi \implies x = \pi
   \]
   So, the surface areas of the two parts are \( 3\pi \) and \( \pi \).

4. **Position of the plane**: The plane must be positioned such that it divides the surface area in the given ratio. This implies that the plane is not passing through the center but at some distance \( d \) from the center of the sphere.

5. **Geometry of the cut**: The plane cuts the sphere into a spherical cap and the remaining part. The surface area of a spherical cap of height \( h \) from the base to the top is given by:
   \[
   A_{\text{cap}} = 2 \pi r h
   \]
   For the given problem, the height \( h \) of the cap must satisfy:
   \[
   2 \pi \cdot 1 \cdot h = 3 \pi \implies h = \frac{3}{2}
   \]
   However, since the height \( h \) cannot be greater than the diameter of the sphere, we need to reconsider the height calculation.

6. **Correct height calculation**: The height \( h \) of the cap should be such that the surface area of the cap is \( 3 \pi \). The height \( h \) from the base to the top of the cap is:
   \[
   h = 1 - d
   \]
   where \( d \) is the distance from the center of the sphere to the plane. The surface area of the cap is:
   \[
   2 \pi (1 - d) = 3 \pi \implies 1 - d = \frac{3}{2} \implies d = -\frac{1}{2}
   \]
   This is not possible since \( d \) must be positive. Therefore, the correct height calculation should be:
   \[
   2 \pi (1 - d) = \pi \implies 1 - d = \frac{1}{2} \implies d = \frac{1}{2}
   \]

7. **Radius of the disk**: The radius \( r \) of the disk cut by the plane can be found using the Pythagorean theorem in the right triangle formed by the radius of the sphere, the distance from the center to the plane, and the radius of the disk:
   \[
   r^2 + \left(\frac{1}{2}\right)^2 = 1^2 \implies r^2 + \frac{1}{4} = 1 \implies r^2 = \frac{3}{4}
   \]
   Therefore, the radius \( r \) of the disk is:
   \[
   r = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}
   \]

8. **Area of the disk**: The area \( A_{\text{disk}} \) of the disk is:
   \[
   A_{\text{disk}} = \pi r^2 = \pi \left(\frac{\sqrt{3}}{2}\right)^2 = \pi \cdot \frac{3}{4} = \frac{3\pi}{4}
   \]

The final answer is \(\boxed{\frac{3\pi}{4}}\)