Step-by-step Solution:

We are given \(|\vec{a}|=3\), \(|\vec{b}|=4\), and \(|\vec{a}+\vec{b}|=1\). We want \(|\vec{a}-\vec{b}|\). Step 1: Use the identity \[ |\vec{a}+\vec{b}|^{2} = |\vec{a}|^{2} + |\vec{b}|^{2} + 2\vec{a}\cdot\vec{b}. \] Substitute values: \[ 1^{2} = 3^{2} + 4^{2} + 2\vec{a}\cdot\vec{b}, \] \[ 1 = 9 + 16 + 2\vec{a}\cdot\vec{b}, \] \[ 2\vec{a}\cdot\vec{b} = 1 - 25 = -24, \] \[ \vec{a}\cdot\vec{b} = -12. \] Step 2: Use the formula for difference \[ |\vec{a}-\vec{b}|^{2} = |\vec{a}|^{2} + |\vec{b}|^{2} - 2\vec{a}\cdot\vec{b}. \] Substitute: \[ |\vec{a}-\vec{b}|^{2} = 9 + 16 - 2(-12), \] \[ = 25 + 24 = 49. \] Final Answer: \[ |\vec{a}-\vec{b}| = \sqrt{49} = 7. \] \[ \boxed{7} \]