Let M be a Turing Machine with a doubly infinite tape. Let N be a Turing Machine with a singly infinite tape. 
We want to show that M can simulate N and N can simulate M.

Case 1: M simulating N

Run M with the same settings as N. Any DFA that works on N also works on M

Case 2: N simulating M

Let the DFA that controls M be in control of N.
Change the DFA such that when it tries to go farther left than it is able to it shifts
all the characters on the tape to the right one space and then returns itself to the first spot on the tape.

This change will allow N to simulate M.