The Ramsey number \(R(P_n, K_1 + P_m)\) is determined for various ranges of \(n\) and \(m\), where \(P_n\) is a path with \(n\) vertices, and \(K_1 + P_m\) is the graph obtained from a path \(P_m\) by adding an additional vertex that is adjacent to all of the vertices of the path. For example, it is shown that \(R(P_n, K_1 + P_m) = 2n - 1\) for \(n \geq 4\), \(m\) even, and \(4 \leq m \leq n + 1\). This result follows a previous paper of the authors on \(R(P_n, W_m)\), where \(W_m\) is the wheel with \(m + 1\) vertices. On certain intervals bounds are given of which the following is an example. If \(n \geq 6\) and \(m\) is even with \(n + 2 \leq m \leq \lfloor n/3 \rfloor\), then \(m + \lfloor (3n)/2 \rfloor - 2 \geq R(P_n, K_1 + P_m) \geq 2n - 1\). Many small order Ramsey numbers are determined for the pair \((P_n, K_1 + P_m)\).