pcm.dsp0 {
    type plug
    slave.pcm "dmix"   # <-- this line has been altered!
}

# mixer0 can stay unchanged, because it isn't used anyway, I guess ;)
ctl.mixer0 {
    type hw
    card 0
}

# I also had to add/alter the following, making ALSA use dmix by default
pcm.!default {
    type plug
    slave.pcm "dmix"
}