I figured out the problem, and I'm posting it here in case it's helpful to anybody else: 

The Qualcomm Atheros network adapter is mounted just above the connector on the mainboard, and this network adapter prevents the latch on the mainboard connector from fully opening.  If the latch isn't fully open, the cable connector will not insert completely into the mainboard connector.  So, I removed the screw holding the network adapter so that the adapter sprang up and out of the way.  Once it was out of the way, it was relatively easy fully open the latch, fully insert the blue end of the hard disk cable into the mainboard connector, and close the latch.  The latch now holds the cable securely in place, and everything is working great.

Thanks very much JackE.  That is the cable I already have, but I cannot figure out how the blue end fits into the motherboard connector.  There's a tutorial on the Acer website (link below) that simply says, "Connect the hard drive cable to the mainboard connector and close the latch."  However, the blue connector on the cable does not really "connect" in any positive way to the mainboard connector.  At best, all I can do is open the latch and just lay the cable end down loosely on top of the mainboard connector, without a positive fit or anything to assure alignment, and try to close the latch.  However, the latch doesn't hold it in place, and the cable simply falls out. What am I doing wrong? 

Here's the link to the tutorial: https://community.acer.com/en/kb/articles/13999-hard-drive-installation-instructions-for-the-acer-aspire-a5-series  Step 10 of the tutorial is what I copied and pasted above.

Thanks very much for your reply!