For a simple connected graph $G=(V,E)$, let $d(u)$ be the degree of the vertex $u$ of $G$. The general Sombor index of $G$ is defined as $$SO_α(G)=\sum_{uv\in E} \left[d(u)^2+d(v)^2\right]^α$$ where $SO(G)=SO_{0.5}(G)$ is the recently invented Sombor index. In this paper, we show that in the class of connected graphs with a fixed degree sequence (for which the minimum degree being equal to one), there exists a special extremal $BFS$-graph with minimum general Sombor index for $0<α<1$ (resp. maximum general Sombor index for either $α>1$ or $α<0$). Moreover, for any given tree, unicyclic, and bicyclic degree sequences with minimum degree 1, there exists a unique extremal $BFS$-graph with minimum general Sombor index for $0<α<1$ and maximum general Sombor index for either $α>1$ or $α<0$.