@Zelazny7's answer works, but if you want to keep ties you could do:
df[which(df$Amount == min(df$Amount)), ]
For example with the following data frame:
df <- data.frame(Name = c("A", "B", "C", "D", "E"),
Amount = c(150, 120, 175, 160, 120))
df[which.min(df$Amount), ]
# Name Amount
# 2 B 120
df[which(df$Amount == min(df$Amount)), ]
# Name Amount
# 2 B 120
# 5 E 120
Edit: If there are NAs in the Amount
column you can do:
df[which(df$Amount == min(df$Amount, na.rm = TRUE)), ]