The Apple-accepted cure for a dead RAM slot is a new logic board. Unless you ahve AppleCare, that will be an expensive fix.
An Apple service option is the "depot repair." Instead of working on your computer in the Apple Store, they send it to a centralized refurbishment facility where they fix anything they find wrong (except for cosmetic damage and damage from abuse like dropping) for a flat fee of a little over US$300. That is between 30 and 40 percent of the parts and labor charges if Apple installs a new board in an Apple Store. Sometimes they will install an new battery at the depot without increasing the flat fee.
The downside is that it takes about a week; for that level of cost savings, I could do without my computer for a week.
So take it to Apple and ask about a depot repair.
A low-tech and low-cost option is to expand on what you are doing now--using one slot. Although the official RAM limit on your model is 8GB RAM (two 4GB models), your model can actually handle a total of 16GB (two 8GB modules. You can ignore the dead slot and put an 8GB module in the good slot. My 2012 MBP has 8GB RAM and is completely happy running Mavericks.
Be sure to ONLY get the RAM from an Mac-savvy vendor. For over a decade I've bought all our Mac RAM from one supplier. This is their page for an 8GB module for your MBP model:
http://eshop.macsales.com/item/OWC/1600DDR3S8GB/
I would also trust Crucial on this one. For a change they recognize that your model can use more RAM than stated in Apple's outdated specs:
http://www.crucial.com/usa/en/compatible-upgrade-for/Apple/macbook-pro-%2813-inc h-and-15-inch%2C-mid-2012%29
