bash: how to intercept every command

Is there a way to intercept every command given to bash? I can intercept a particular command, e.g., cd by defining a function cd() and I can do that for one-command-at-a-time for other commands as well. But can I write a function which gets called before every command is executed? I want to do some bookkeeping of commands, and then executed the command.

Michał Šrajer's idea PS4='$(echo $(date) $(history 1) >> /tmp/trace.txt) TRACE: ' looks very promising but here is the output I get:

$ ping www.google.com
 TRACE: ping www.google.com
PING www.l.google.com (74.125.224.52) 56(84) bytes of data.
64 bytes from 74.125.224.52: icmp_seq=1 ttl=56 time=3.77 ms
64 bytes from 74.125.224.52: icmp_seq=2 ttl=56 time=2.33 ms
^C
--- www.l.google.com ping statistics ---
2 packets transmitted, 2 received, 0% packet loss, time 1001ms
rtt min/avg/max/mdev = 2.334/3.054/3.774/0.720 ms
  TRACE: echo -ne '\033]0;myhost.com /home/yogeshwer/github/myproject\007'
  TRACE: grep -e '\* '
  TRACE: git branch
  TRACE: sed 's/^..\(.*\)/ {\1}/'

And Wed Aug 3 12:47:27 PDT 2011 6672 ping www.google.com get written in /tmp/trace.txt exactly 5 times. The four other ones comes from my definition of PS1 where I run the following command: $(git branch 2> /dev/null | grep -e "\* " | sed "s/^..\(.*\)/ {\1}/"). Two questions:

• Is it possible to write the command to /tmp/trace.txt exactly?
• And more importantly, is it possible to not clutter the output of the command, but only write the command to /tmp/trace.txt?

I am so excited about the possibility of being able to record commands from all my bash sessions in one place!