Bulk Insert With Only One Column

Hi,
I have a question on inserting data for only a specific column in a table.
I have a table as follows
Table <MyTable>
{

Name varchar,
DateUpdate DateTime
}

I wanted to insert the the from a file to table.
The File contains the list of name as follows (line by line)

name1
name2
name3
name4
......


The file name actually contains the DateTime
I would like to insert the names in the file as wellas the DateTime (i.e. part of the file name ) into <MyTable>

I guess "Bulk insert " doens't allow to insert values for only one column

If i change the contents of my data file to

name1 | DateTime1
name2 | DateTime2
name3 | DateTime3
name4 | DateTime4

Then the follwoin query works fine for me.

Bulk Insert <MyTable> FROM <filePath>
With
{


FIELDTERMINATOR = '|'

ROWTERMINATOR = '
'
}

But my original file will contains only Names and the file name contains the date that commom for all the names in the file. And also the file may contains millions of names

Is there any way this can be accomplised using " Bulk Insert" ? Or is there any alternative that i can do it fastly

Your answer will be appreciated

~mohan