Determine the volume of the body obtained by cutting the ball of radius $R$ by the trihedron with vertex in the center of that ball, it its dihedral angles are $\alpha, \beta, \gamma.$
1. **Understanding the Problem:**
   We need to find the volume of the body obtained by cutting a sphere of radius \( R \) with a trihedron whose vertex is at the center of the sphere and whose dihedral angles are \( \alpha, \beta, \gamma \).

2. **Volume Ratio Approach:**
   The volume of the sphere is given by:
   \[
   V_{\text{sphere}} = \frac{4}{3} \pi R^3
   \]
   The volume of the body obtained by the trihedron can be found by considering the ratio of the area of the spherical triangle (formed by the trihedron on the surface of the sphere) to the total surface area of the sphere.

3. **Surface Area of the Sphere:**
   The surface area of the sphere is:
   \[
   A_{\text{sphere}} = 4 \pi R^2
   \]

4. **Area of the Spherical Triangle:**
   The area of a spherical triangle with angles \( \alpha, \beta, \gamma \) on a sphere of radius \( R \) is given by:
   \[
   A_{\text{spherical triangle}} = (\alpha + \beta + \gamma - \pi) R^2
   \]
   This formula comes from the spherical excess theorem, which states that the area of a spherical triangle is proportional to its spherical excess (the sum of its angles minus \(\pi\)).

5. **Volume Calculation:**
   The ratio of the area of the spherical triangle to the total surface area of the sphere is:
   \[
   \frac{A_{\text{spherical triangle}}}{A_{\text{sphere}}} = \frac{(\alpha + \beta + \gamma - \pi) R^2}{4 \pi R^2} = \frac{\alpha + \beta + \gamma - \pi}{4 \pi}
   \]
   Therefore, the volume of the body obtained by the trihedron is:
   \[
   V_{\text{body}} = \left( \frac{\alpha + \beta + \gamma - \pi}{4 \pi} \right) \cdot V_{\text{sphere}}
   \]
   Substituting the volume of the sphere:
   \[
   V_{\text{body}} = \left( \frac{\alpha + \beta + \gamma - \pi}{4 \pi} \right) \cdot \frac{4}{3} \pi R^3
   \]
   Simplifying this expression:
   \[
   V_{\text{body}} = \frac{(\alpha + \beta + \gamma - \pi) R^3}{3}
   \]

The final answer is \( \boxed{ \frac{(\alpha + \beta + \gamma - \pi) R^3}{3} } \)