Here is my plot
dat <- data.frame(
pos = c(1, 3, 5, 8, 10, 12),
start = c(1,3, 6, 7, 10, 11),
end = c(5, 6, 9, 9, 13, 12)
)
library(ggplot2)
p <- ggplot(dat) + geom_segment(aes(x=start, y=pos, xend=end, yend=pos),
color="blue", size=2) + ylab("Fragments)") + xlab("Position")
scale_y_reverse() + theme_bw()
p1 <- p + opts(legend.position="left",
panel.background=theme_blank(),panel.border=theme_blank(),
panel.grid.major=theme_blank(),
panel.grid.minor=theme_blank(),plot.background=theme_blank())
p1

Bitmapped desired version is, with axis line and labels near the segments. [ just exta-note: Note that bitmap changed the line to round ended (it would be interesting to see if we can do in the ggplot2)]

Interestingly, I actually think this is easier in base graphics:
Edit Added additional
axiscall to get the outer most tick in both directions.