This week's book giveaway is in the OCAJP 8 forum. We're giving away four copies of OCA Java SE 8 Programmer I Study Guide and have Edward Finegan & Robert Liguori on-line! See this thread for details.
FileOutputStream and FileWriter both have constructors which allow you to specify a boolean argument. If set to true you open in "append" mode, so newly-written bytes are added to the end of an existing file. (Otherwise the default is to overwrite any existing file.) So you can copy the first file to its new location (or leave it in place if you prefer.) Then read from the second file, and use a Writer or OutputStream in append mode to add what you've read to the first file.