Let \(G\) be a graph of order \(n \geq 4k+8\), where \(k\) is a positive integer with \(kn\) even and \(\delta(G) > k+1\). We show that if \(max\{d_G(u),d_G(v)\} > {n}/{2}\) for each pair of nonadjacent vertices \(u,v\), then \(G\) has a connected \([k, k+1]\)-factor excluding any given edge \(e\).